Applied Science, Faculty of
Mechanical Engineering, Department of
DSpace
UBCV
Yang, Shenliang
Doctor of Philosophy - PhD
University of British Columbia
As a widely used metal additive manufacturing technology, directed energy deposition (DED) provides a cost-effective solution for repairing and remanufacturing high-performance metallic components. Nevertheless, their industrial applications are limited by excessive geometric distortion (caused by high-gradient bulk residual stress) and poor surface quality. To achieve desired surface finishes, post-machining of DED is usually performed, which introduces additional surface residual stress. These residual stresses can compromise structural integrity, fatigue life, and corrosion resistance. Guided by three critical and practical research questionsâ(1) how to efficiently predict DED-induced bulk residual stress, (2) how to accurately simulate surface residual stress created by machining, and (3) how DED-induced bulk stress interacts with machining to generate final surface stressâthis thesis develops physics-based models to understand the formation mechanism of residual stresses in combined DED and machining processes. First, analytical models are proposed to determine thermo-mechanical responses in single-material and functionally graded DED processes. These models employ finite difference methods for predicting temperature cycles, modified Greenâs functions to quantify thermal stress, and radial return methods to update plastic stress, which features the material addition as the heat source moves in each deposition layer in DED. The frameworks effectively demonstrate how DED deposition conditions and material gradients influence bulk residual stress. Second, finite element method (FEM) models are presented to predict machining-induced surface residual stress in conventionally wet machining processes and high-speed machining processes with serrated chip generation. The results quantitatively illustrate the mitigation of surface tensile residual stress due to applying cutting fluids and the fluctuation of surface residual stress caused by periodic shear localization in chips. Lastly, the FEM machining model is extended by incorporating initial stress states and microstructure features to simulate post-machining DED processes. Results suggest that pre-existing residual stress significantly influences final residual stress, while the finer the grain size the more compressive residual stress on the post-machined surfaces. By revealing the formation mechanisms of residual stresses in DED and machining processes, this thesis offers practical guidance for planning the combined additive manufacturing and post-machining technologies, thereby fostering the adoption of advanced manufacturing technologies in mission-critical industries.
Modelling of Residual Stresses inCombined Additive Manufacturingand Machining ProcessesbyShenliang YangA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)May 2025© Shenliang Yang 2025The following individuals certify that they have read, and recommend to the Faculty of Graduateand Postdoctoral Studies for acceptance, the dissertation entitled:Modelling of Residual Stresses in Combined Additive Manufacturing and Machining Processessubmitted by Shenliang Yang in partial fulfilment of the requirements forthe degree of Doctor of Philosophyin Mechanical EngineeringExamining Committee:Xiaoliang Jin, Associate Professor, Mechanical Engineering, UBCSupervisorAdam Clare, Professor, Mechanical Engineering, UBCCo-supervisorRajeev Jaiman, Professor, Mechanical Engineering, UBCUniversity ExaminerEdouard Asselin, Professor, Materials Engineering, UBCUniversity ExaminerIsmail LazogËlu, Professor, Mechanical Engineering, Koc UniversityExternal ExaminerAdditional Supervisory Committee Members:Yusuf Altintas, Professor, Mechanical Engineering, UBCSupervisory Committee MemberGary Schajer, Professor, Mechanical Engineering, UBCSupervisory Committee MemberChadwick Sinclair, Professor, Materials Engineering, UBCSupervisory Committee MemberiiAbstractAs a widely used metal additive manufacturing technology, directed energy deposition (DED) pro-vides a cost-effective solution for repairing and remanufacturing high-performance metallic com-ponents. Nevertheless, their industrial applications are limited by excessive geometric distortion(caused by high-gradient bulk residual stress) and poor surface quality. To achieve desired surfacefinishes, post-machining of DED is usually performed, which introduces additional surface resid-ual stress. These residual stresses can compromise structural integrity, fatigue life, and corrosionresistance.Guided by three critical and practical research questionsâ(1) how to efficiently predict DED-induced bulk residual stress, (2) how to accurately simulate surface residual stress created bymachining, and (3) how DED-induced bulk stress interacts with machining to generate final sur-face stressâthis thesis develops physics-based models to understand the formation mechanism ofresidual stresses in combined DED and machining processes.First, analytical models are proposed to determine thermo-mechanical responses in single-material and functionally graded DED processes. These models employ finite difference meth-ods for predicting temperature cycles, modified Greenâs functions to quantify thermal stress, andradial return methods to update plastic stress, which features the material addition as the heatsource moves in each deposition layer in DED. The frameworks effectively demonstrate how DEDdeposition conditions and material gradients influence bulk residual stress.Second, finite element method (FEM) models are presented to predict machining-induced sur-face residual stress in conventionally wet machining processes and high-speed machining processeswith serrated chip generation. The results quantitatively illustrate the mitigation of surface tensileresidual stress due to applying cutting fluids and the fluctuation of surface residual stress causedby periodic shear localization in chips.Lastly, the FEM machining model is extended by incorporating initial stress states and mi-crostructure features to simulate post-machining DED processes. Results suggest that pre-existingresidual stress significantly influences final residual stress, while the finer the grain size the morecompressive residual stress on the post-machined surfaces.By revealing the formation mechanisms of residual stresses in DED and machining processes,this thesis offers practical guidance for planning the combined additive manufacturing and post-machining technologies, thereby fostering the adoption of advanced manufacturing technologies inmission-critical industries.iiiLay SummaryRemanufacturing is a sustainable solution for expanding the lifespan of high-performance metalliccomponents, particularly in the aerospace and energy industries. As an advanced metal additivemanufacturing, directed energy deposition (DED) is widely employed to rebuild or restore damagedparts. Meanwhile, to ensure satisfactory party geometry and surface quality, machining is usuallyperformed after DED. However, both DED and machining introduce complex residual stresses,which can cause distortions, weaken corrosion resistance, and shorten the service life of criticalparts.This thesis develops predictive models to simulate how residual stresses form and interact duringDED and machining processes. DED models show how heat transfer and material addition generatebulk stress, while machining models demonstrate how cutting conditions and initial states influencefinal surface stress. These models allow engineers to customize process parameters, minimize defectscaused by residual stresses, and enhance the reliability of manufactured components, especially forremanufacturing applications.ivPrefaceThis thesis was written to fulfill the requirement of the degree of Doctor of Philosophy in MechanicalEngineering at the University of British Columbia. The generative artificial intelligence tools wereonly used for grammar and spelling correction in the preparation of this thesis.Under the supervision of Dr. Xiaoliang Jin and Dr. Adam Clare, the author, Shenliang Yang,developed all predictive models, including analytical models to predict the bulk residual stress dis-tribution in additively manufactured structures and numerical models to simulate surface residualstress profiles after machining. Meanwhile, the author designed and conducted the majority of theexperiments to validate these models, including all the DED and machining experiments, metallur-gical sample preparations, as well as optical, laser, and scanning electron microscopy observationsand characterizations.Due to the resource limitation, several steps were performed by our collaborators or obtainedcommercially, including residual stresses on FGM samples (from Lambda Technologies Group.),Waspaloy samples (from Pratt & Whitney), and single-track IN718 samples (from the Universityof Nottingham).A version of Chapter 2 has been published in [Yang, S.], Clare, A.T., Bennett, C. and Jin,X., 2024, Informing Directed Energy Deposition Strategies through Understanding the Evolutionof Residual Stress, Additive Manufacturing, 79, 103907. The contributions of the authors arespecified as follows. Yang, S.: Writing â original draft, Visualization, Validation, Methodology,Investigation, Formal analysis, Data curation, Conceptualization. Clare, A.T.: Writing â review &editing, Supervision, Methodology, Investigation, Formal analysis, Conceptualization. Bennett, C.:Writing â review & editing, Methodology, Investigation, Formal analysis, Conceptualization. Jin,X.: Writing â review & editing, Supervision, Project administration, Methodology, Investigation,Funding acquisition, Formal analysis, Conceptualization.A version of Chapter 3 has been published in [Yang, S.], Clare, A.T. and Jin, X., 2024, Un-derstanding Residual Stress in Functionally Graded Directed Energy Deposition, Additive Man-ufacturing, 96, 104581. The specific contributions of each author are detailed below. Yang, S.:Writing â original draft, Visualization, Validation, Software, Methodology, Investigation, Formalanalysis, Data curation, Conceptualization. Clare, A.T.: Writing â review & editing, Supervision,Methodology, Investigation, Formal analysis, Conceptualization. Jin X.: Writing â review & edit-ing, Supervision, Project administration, Methodology, Investigation, Funding acquisition, Formalanalysis, Conceptualization.A version of Chapter 4 has been published in [Yang, S.], Jin, X., Engin, S., Kountanya, R., El-vPrefaceWardany, T. and Lee, S.Y., 2023, Effect of Cutting Fluids on Surface Residual Stress in Machining ofWaspaloy, Journal of Materials Processing Technology, 322, 118170. The authorship contributionsare clarified as follows. Yang, S.: Conceptualization, Methodology, Validation, Formal analysis,Investigation, Data curation, Writing â original draft, Visualization. Jin, X.: Conceptualization,Methodology, Formal analysis, Investigation, Resources, Writing â review & editing, Supervision,Project administration, Funding acquisition. Engin, S.: Methodology, Formal analysis, Investiga-tion. Kountanya, R.: Methodology, Formal analysis, Investigation, Resources. El-Wardany, T.:Methodology, Formal analysis, Investigation. Lee, S.Y.: Formal analysis, Investigation, Validation.A version of Chapter 5 has been published in [Yang, S.], Jin, X., Engin, S., Kountanya, R.and El-Wardany, T., 2024, Effect of Shear Localization on Surface Residual Stress Distribution inMachining of Waspaloy, ASME Journal of Manufacturing Science and Engineering, 146(11). Thecontributions of the authors are outlined as follows. Yang, S.: Conceptualization, Methodology,Validation, Formal analysis, Investigation, Data curation, Writing â original draft, Visualization.Jin, X.: Conceptualization, Methodology, Formal analysis, Investigation, Resources, Writing âreview & editing, Supervision, Project administration, Funding acquisition. Engin, S.: Methodol-ogy, Formal analysis, Investigation. Kountanya, R.: Methodology, Formal analysis, Investigation.El-Wardany, T.: Methodology, Formal analysis, Investigation.A version of Chapter 6 has been published in: [Yang, S.], Speidel, A., Clare, A.T., Bennett, C.and Jin, X., Surface Residual Stress Following Machining of Directed Energy Deposition Structures,Additive Manufacturing, 104, 104765. The contributions of the authors are described as follows.Yang, S.: Writing â original draft, Visualization, Validation, Methodology, Investigation, Formalanalysis, Data curation, Conceptualization. Speidel, A.: Writing â review & editing, Methodology,Formal analysis, Investigation, Validation. Clare, A.T.: Writing â review & editing, Supervision,Methodology, Investigation, Formal analysis, Conceptualization. Bennett, C.: Methodology, For-mal analysis, Investigation. Jin X.: Writing â review & editing, Supervision, Project administration,Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxviiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxxi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Thesis Objectives and Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Bulk Residual Stress in Single-Material Directed Energy Deposition Structures 62.1 Introduction and Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Single-Material DED Process Modelling and Validation . . . . . . . . . . . . . . . . 102.2.1 Prediction of temperature field . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Determination of bulk residual stress . . . . . . . . . . . . . . . . . . . . . . 202.2.3 Calculation of geometric distortion . . . . . . . . . . . . . . . . . . . . . . . 322.3 Analyses on Stress Evolution in Single-Material DED Processes . . . . . . . . . . . 352.4 Guidance for Single-Material DED Strategies . . . . . . . . . . . . . . . . . . . . . . 392.4.1 Linear energy density parameters . . . . . . . . . . . . . . . . . . . . . . . . 392.4.2 DED process planning parameters . . . . . . . . . . . . . . . . . . . . . . . . 412.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43vii3 Bulk Residual Stress in Functionally Graded Directed Energy Deposition Struc-tures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.1 Introduction and Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 Modelling of Bulk Residual Stress in DED-built FGM Structures . . . . . . . . . . . 473.2.1 Modelling principles and assumptions . . . . . . . . . . . . . . . . . . . . . . 473.2.2 Evaluating material properties in transitional regions . . . . . . . . . . . . . 473.2.3 Bulk residual stress modelling procedure . . . . . . . . . . . . . . . . . . . . 483.3 DED of FGM Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . 523.3.1 Fabrication conditions of FGM structures . . . . . . . . . . . . . . . . . . . . 523.3.2 EDS tests and analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.3.3 Residual stress measurements and predictions . . . . . . . . . . . . . . . . . 603.4 Bulk Residual Stress in DED-built FGM Structures Featuring Adapted TransitionalRegions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684 Surface Residual Stress in Wet Machining Processes . . . . . . . . . . . . . . . . 704.1 Introduction and Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.2 Wet Machining Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2.1 Orthogonal cutting model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2.2 Cooling down model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.3 Experimental Setup and Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.4.1 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.4.2 Effect of cutting fluid on local normal/shear contact stresses . . . . . . . . . 934.4.3 Effect of cutting fluid on machined surface temperature distribution . . . . . 954.4.4 Effect of cutting fluid on machined surface residual stress . . . . . . . . . . . 994.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025 Surface Residual Stress in High-Speed Machining Processes with Serrated ChipFormation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.1 Introduction and Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.2 Model Description and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.2.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.2.2 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.3.1 Tool-workpiece contact forces . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.3.2 Machined surface topography . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.3.3 Machined surface residual stress and plastic strain . . . . . . . . . . . . . . . 1165.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119viiiTable of Contents6 Surface Residual Stress Following Machining of Directed Energy DepositionStructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.1 Introduction and Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.2 Post-Machining DED Model Description . . . . . . . . . . . . . . . . . . . . . . . . 1236.2.1 Initial geometries, meshing strategies, and kinematics . . . . . . . . . . . . . 1256.2.2 Material models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.3 Experiments and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.3.1 DED experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.3.2 Post-machining experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.3.3 Residual stress measurements and reconstruction . . . . . . . . . . . . . . . 1396.3.4 Grain size measurements and analysis . . . . . . . . . . . . . . . . . . . . . . 1416.3.5 Model validation and discussions . . . . . . . . . . . . . . . . . . . . . . . . . 1446.4 Quantitative Analyses of Surface Residual Stress after Post-Machining DED . . . . 1476.4.1 Cutting forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.4.2 Final surface residual stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1546.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1617 Conclusions and Future Research Directions . . . . . . . . . . . . . . . . . . . . . 163Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167AppendicesA Determination of Cell Size and Time Increment in Solving Heat Transfer Equa-tions through Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . 184B The Necessity to Include Surface Normal Tractions and Hydrostatic Stresses inDED-Induced Bulk Residual Stress Calculation . . . . . . . . . . . . . . . . . . . 185C Estimating Relaxed Bulk Residual Stress in DED-Fabricated FGM StructuresDue to Material Removal through WEDM and Electropolishing . . . . . . . . . 187D Quantifying Relaxed Residual Stress on Machined Surfaces Caused by Separat-ing Specimens Using WEDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190E Characterization of Stress States by Stress Triaxiality and Lode Parameter . . 192ixList of Tables2.1 Overview of FEM-based models used for predicting the bulk residual stress in single-material DED structures reported in the literature. The evaluation of their relativestrengths and weaknesses is a key motivation for the present work. . . . . . . . . . . 82.2 DED process parameters for the validation of temperature [96, 123]. . . . . . . . . . 172.3 Peak temperature comparison and their prediction error. (Units: K) . . . . . . . . . 192.4 Mean absolute temperature error in each cycle compared to the measurements in [96]. 192.5 Single-material DED process parameters for the validation of bulk residual stress [41]. 292.6 Material properties of mild steel S315 used for the model presented here. . . . . . . . 302.7 JC plasticity constants for mild steel S315 [41]. . . . . . . . . . . . . . . . . . . . . . 302.8 Comparison of computational time between the FEM-based model in [41] and thecurrent model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.9 Comparison of bulk residual stress indicators and distortion under three depositioncases, illustrating the utility of this modelling approach for appraising and ultimatelyoptimization of deposition parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 413.1 DED process conditions for fabricating the FGM thin-wall structures. . . . . . . . . 533.2 Four criteria and their corresponding upper/lower boundaries to determine actualweight percentages of SS316 and IN718 materials in each deposition layer. (Units:wt%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.1 Material properties of cutting tool (uncoated cemented carbide), workpiece (Was-paloy), and cutting fluid (water). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2 Coefficients of the JC plasticity model for Waspaloy. . . . . . . . . . . . . . . . . . . 764.3 Coefficients of the JC damage initiation model for Waspaloy. . . . . . . . . . . . . . 774.4 Cutting tool geometric parameters used in the experiments. . . . . . . . . . . . . . . 844.5 Experimental conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.6 Surface residual stress measurement conditions. . . . . . . . . . . . . . . . . . . . . . 875.1 Overview of FEM-based models used for the prediction of machining-induced residualstress profiles in orthogonal cutting of high-strength metallic alloys. . . . . . . . . . . 1055.2 Orthogonal cutting conditions for model validations. . . . . . . . . . . . . . . . . . . 1095.3 Comparison of chip geometries between simulations and experimental measurementsin [46] for cutting conditions L and H . . . . . . . . . . . . . . . . . . . . . . . . . 110xList of Tables6.1 Coefficients of the JC plasticity model for DED-built IN718 [14, 50, 165]. . . . . . . 1286.2 Coefficients of the JC damage initiation model for IN718 [44, 154, 174]. . . . . . . . 1326.3 Orthogonal cutting parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.4 Details of residual stress measurement conditions for DED-built single-track IN718samples after orthogonal cutting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.5 dg x and dg y values in each plane and their corresponding mean values. . . . . . . . 1446.6 Comparison between measured and simulated cutting forces under Vc = 15 m/minand H = 0.12 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1456.7 Details about 9 post-machining simulations of IN718. . . . . . . . . . . . . . . . . . . 149xiList of Figures1.1 Representative procedures to restore a damaged turbine blade through DED + ma-chining processes: (a) Original/reconstructed blade; (b) Damaged blade; (c) Frac-ture surface after pre-machining; (d) Deposited area by DED; (e) Repaired area aftermachining. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Failures of metallic components caused by manufacturing-induced residual stresses:(a) Dimension inaccuracy [110]; (b) Cracking [130]; (c) Delamination [36]; (d) De-formation [110, 113]; (e) Stress corrosion [71]. These figures are reprinted withpermission from Springer Nature and Elsevier. . . . . . . . . . . . . . . . . . . . . . 21.3 Graphical outline of this thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Temperature and stress distribution during laser-based powder-fed single-materialDED processes along (a) Path 1, (b) Path 2, (c) Path 3. x, y and z denote heatsource traverse (longitudinal), transverse, and build (vertical) directions, respec-tively. Paths 1, 2, and 3 represent intersection lines between the top surface of thesubstrate and planes 1, 2, and 3, respectively. Points oi (i = 1, 2, and 3) are themiddle points of the three lines, and oiy i (i = 1, 2, and 3) refers to their positivedirections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 An approach for modelling temperature, bulk residual stress and geometric distortionof DED-built single-material thin-wall structures. Sequentially coupled temperature,thermal stress and residual stress are calculated. The process maybe repeated asthermal âloadingâ and/or âunloadingâ takes place in subsequent layers of deposition.In the end, the obtained bulk residual stress profile is used to calculate the geometricdistortion of the structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Schematics of temperature modelling in single-material DED processes: (a) 2D com-putation domain and boundary conditions at a deposition time of t ; (b) The finitedifference method (FDM) used to update temperature. . . . . . . . . . . . . . . . . . 132.4 The change of boundary conditions at a cell of interest during a DED process. Bound-ary conditions are subject to change as the modes of heat transfer available to themalso change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 Comparison of the measured thermal cycles in [96] with respect to the predictionsby the FVM-based model in [123] and calculations by the current FDM-based model. 19xiiList of Figures2.6 A schematic diagram of the thermal stress calculation at point (x, z ): (a) Bodyforces; (b) Surface normal traction; (c) Hydrostatic stress. . . . . . . . . . . . . . . . 212.7 Comparison of longitudinal residual stress in DED-built single-material thin-wallstructures between the FEM-based model [41] and the current model. . . . . . . . . 312.8 Calculation of the equivalent bending moment caused by DED-induced bulk residualstress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.9 Comparison between measured [41] and our predicted distortion profiles for a DED-fabricated single-material thin-wall structure. . . . . . . . . . . . . . . . . . . . . . . 342.10 Evolution of temperature/stress/strain at selected points during the fabrication ofthe first layer for a single-material DED thin-wall structure. The deposition con-ditions are the same as the residual stress validation in Section 2.2.2. When theheat source moves into the mid-length of the deposit, the three points are located at(a) the new-added zone, (b) the upper heated zone, and (c) the lower heated zone,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.11 Evolution of temperature/stress/strain at (a) point 1, (b) point 2, and (c) point 3 inthe fabrication of the second layer for a DED-built single-material thin-wall structure. 382.12 Effect of Pl and vl on (a) surface residual stress, (b) change of the average residualstress in deposits, (c) stress reverse position, and (d) maximum vertical distortion.(e) shows a typically deformed single-material thin-wall structure fabricated by DED. 402.13 Effect of tidle and deposition pattern on (a) surface residual stress, (b) change ofthe average residual stress in deposits, (c) stress reverse position, and (d) maximumvertical distortion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.1 Modelling of temperature and bulk residual stress in DED-built FGM thin-wall struc-ture. The thermal stress loading process occurs at each time increment ât. Ï1 andÏ2 refer to the principal stress components. . . . . . . . . . . . . . . . . . . . . . . . 493.2 Experimental setup, deposition strategy, and metal powder information used in thefabrication of DED-built SS316/IN718 FGM thin-wall structures. In the Cartesiancoordinate system O-XYZ, the X, Y , and Z axes are parallel to the longitudinaldirection, transverse direction, and build direction, respectively. . . . . . . . . . . . . 533.3 SEM images, energy spectra, and measured element weight ratios at the transversecross-section of DED-produced FGM thin-wall structures for the selected areas at(a) the substrate, (b) layer 3, (c) layer 10, (d) layer 15, (e) layer 20, and (f) layer 28. 573.4 Measured weight ratio profiles of the elements (a) Fe, (b) Cr, (c) Ni, (d) Mo, (e)Nb, and (f) Ti at the transverse cross-section of FGM thin-wall structures along thebuild direction. The profiles start at the interface between the substrate and theFGM. The measured element weight ratios of SS316 and IN718 metal powders withcertain error bands are plotted to identify the mixed region of the two materials. . . 58xiiiList of Figures3.5 Weight ratios of SS316 and IN718 materials at each deposition layer of the FGMstructure: designed weight ratios (a) and actual weight ratios identified by the fourcriteria based on the measured weight percentage of the elements (b) Fe, (c) Ni, (d)Nb, and (e) Ti. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.6 Residual stress measurements of the FGM structure: (a) WEDM strategies; (b)Electropolishing; (c) Measurement area; (d) Detailed locations of each measurementarea. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.7 Temperature- and composition-dependent material properties used in the bulk resid-ual stress prediction in the fabrication of SS316/IN718 thin-wall structures by DED:(a) Density; (b) Thermal conductivity; (c) Specific heat capacity; (d) Thermal ex-pansion coefficient; (e) Youngâs modulus; (f) Poissonâs ratio. . . . . . . . . . . . . . 623.8 Comparison between measured residual stress points and the simulated bulk residualstress profiles based on (a) designed weight percentages of SS316 and IN718 in eachdeposition layer; and actual weight percentages of the two materials identified by theweight ratios of elements (b) Fe, (c) Ni, (c) Nb, and (d) Ti. . . . . . . . . . . . . . . 653.9 Two DED deposition strategies for fabricating FGM thin-wall structures featuredby titled material composition transition regions: (a) Vertical- and (b) Horizontal-dominant FGM walls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.10 Effects of material transitional angles on the bulk residual stress distributions ofvertical-dominant walls under θ1 = â2.4â¦, θ1 = 0â¦, and θ1 = 2.4â¦, and horizontal-dominant walls under θ2 = 45â¦, θ2 = 90â¦, and θ2 = 135â¦. . . . . . . . . . . . . . . . . 674.1 A schematic diagram of wet cutting processes. . . . . . . . . . . . . . . . . . . . . . . 704.2 A schematic diagram of wet cutting simulation. . . . . . . . . . . . . . . . . . . . . . 734.3 Initial geometry and boundary conditions of the general wet cutting model (H =0.12 mm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.4 Flow states of cutting fluid in wet cutting: (a) Schematic diagram of flowing states;(b) Simulated flowing conditions (Vc = 86 m/min and H = 0.06 mm). . . . . . . . . 804.5 Fluid velocity prediction model and simulation results: (a) Initial geometry andboundary conditions; (b) Resultant velocity distribution (Vf = 6400 mm/s and Ï =120 mm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.6 Simulation results of cutting fluid velocity (a) along Path 1; (b) at the selected point. 824.7 Cooling down model with the application of boundary conditions. . . . . . . . . . . . 824.8 Experimental setup, schematic diagrams of wet cutting process, and tool radiusmeasurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.9 Sample preparations for surface residual stress measurements and microstructureobservations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.10 Cutting forces predicted by the numerical model during wet cutting of Waspaloyunder conditions of Vc = 86 m/min and H = 0.06 mm. . . . . . . . . . . . . . . . . . 88xivList of Figures4.11 Measured and filtered cutting forces (Vc = 86 m/min, H = 0.06 mm, wet cutting):(a) DOC = 0.75 mm; (b) DOC = 1.5 mm. . . . . . . . . . . . . . . . . . . . . . . . . 894.12 Comparison of average tangential (Ft) and feed (Ff ) forces between experiments andsimulations under Vc = 86 m/min (a) H = 0.06 mm; (b) H = 0.12 mm. . . . . . . . 904.13 Surface residual stress distribution in the cutting velocity direction (Vc = 86 m/minand H = 0.06 mm) and selected paths: (a) Dry cutting; (b) Wet cutting. . . . . . . 914.14 Comparison between experimentally and numerically obtained residual stress profilesin cutting velocity direction with Vc = 86 m/min (a) H = 0.06 mm, dry cutting; (b)H = 0.06 mm, wet cutting; (c) H = 0.12 mm, dry cutting; and (d) H = 0.12 mm,wet cutting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.15 Materials flow around the tool tip during orthogonal cutting of Waspaloy: (a) Drycutting; (b) Wet cutting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.16 Contact forces at the tool-workpiece interface (Vc = 86 m/min, H = 0.06 mm, wetcutting): (a) CNORMF; (b) CSHEARF; (c) Contact force evolution at the point S. 944.17 Average local contact stress when orthogonal cutting of Waspaloy under Vc = 86m/min and (a) H = 0.06 mm; (b) H = 0.12 mm. . . . . . . . . . . . . . . . . . . . . 954.18 Temperature distribution during orthogonal cutting processes in the case of Vc = 86m/min, H = 0.06 mm. (a) Dry cutting; (b) Wet cutting. . . . . . . . . . . . . . . . 964.19 Workpiece temperature distribution during orthogonal cutting processes at Vc = 86m/min. (a) Path 1, H = 0.06 mm; (b) Path 2, H = 0.06 mm; (c) Path 1, H = 0.12mm; (d) Path 2, H = 0.12 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.20 Workpiece temperature distribution during cooling down processes in the case of Vc= 86 m/min, H = 0.06 mm. (a) Dry cutting; (b) Wet cutting. . . . . . . . . . . . . 984.21 Workpiece temperature distribution during cooling down processes under Vc = 86m/min. (a) Path 1, H = 0.06 mm; (b) Path 2, H = 0.06 mm; (c) Path 1, H = 0.12mm; (d) Path 2, H = 0.12 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.22 Thermal history, plastic strain, and stress evolution at a surface point during orthog-onal cutting of Waspaloy: (a) Vc = 86 m/min, H = 0.06 mm; (b) Vc = 86 m/min,H = 0.12 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.23 Simulated thermal gradients on machined surfaces when orthogonal cutting Was-paloy: (a) Vc = 86 m/min, H = 0.06 mm; (b) Vc = 86 m/min, H = 0.12 mm. . . . . 1014.24 Surface microstructure after orthogonal cutting Waspaloy with Vc = 86 m/min (a)H = 0.06 mm, dry cutting; (b) H = 0.06 mm, wet cutting; (c) H = 0.12 mm, drycutting; and (d) H = 0.12 mm, wet cutting. . . . . . . . . . . . . . . . . . . . . . . . 1025.1 Initial geometries and boundary conditions of the updated FEM-based model tosimulate (a) orthogonal cutting and (b) cooling down processes; H refers to theuncut chip thickness, which is set as 0.12 mm in (a). . . . . . . . . . . . . . . . . . . 1075.2 Comparisons between measured and predicted chip morphologies for cutting condi-tions L and H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109xvList of Figures5.3 Measured and simulated surface profiles along the cutting velocity direction for cut-ting conditions (a) L and (b) H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.4 Distribution and evolution of contact forces at the tool-workpiece interface duringcontinuous chip formation (case L ): Contact forces distribution in (a) normal and(b) shear directions at t = 0.43 ms; Contact forces evolution at (c) Point A, (d) PointS, and (e) Point B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.5 Distribution and evolution of contact forces at the tool-workpiece interface duringserrated chip formation (case H ): Contact forces distribution in (a) normal and (b)shear directions at t = 0.43 ms; Contact forces evolution at (c) Point A, (d) PointS, and (e) Point B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.6 Effects of chip morphologies on the distribution of shear contact forces and stagnationangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.7 Comparisons of machined surface topographies at cutting conditions (a) L and (b)H when orthogonal cutting of Waspaloy. . . . . . . . . . . . . . . . . . . . . . . . . 1155.8 Relationship between stagnation angles and machined surface topographies at cut-ting conditions L and H : (a) Evolution curves of stagnation angle; (b) Detaileddescription of the generation of machined surface topographies. . . . . . . . . . . . . 1165.9 Surface residual stress distribution at cutting conditions (a) L and (b) H ; (c)Comparison of residual stress profiles on the machined surfaces between cuttingconditions L and H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.10 The distribution of normal contact stresses on the cutting tool when t is 0.455 msunder the cutting conditions (a) L and (b) H ; The evolution of the sum of (c)CNORMF and (d) CSHEARF below the stagnation point during simulations underthe above two cutting conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.11 Distribution of the equivalent plastic strain under the cutting conditions (a) L and(b) H ; (c) Comparison of the equivalent plastic strain curves at the finished surfacesunder the above two cutting conditions. . . . . . . . . . . . . . . . . . . . . . . . . . 1196.1 A schematic diagram of post-machining DED simulation showing the interactionbetween consecutive processes and their respective stress conditions. . . . . . . . . . 1246.2 Initial geometry and boundary conditions of the FEM-based numerical modellingprocedure to predict the final residual stress after post-machining of DED-builtsingle-track IN718 samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.3 Comparison of the flow stress-strain curves between JC and MJC laws at a materialpoint under different initial stress states and having (a) dg x = 10 µm and (b) dg x= 100 µm. (c) shows a comparison of the key features in flow stress curves underthe given initial stress states and grain size values. Under the given initial tensileand free-of-stress states, the stress triaxiality term (Term 5) yields the same valueof 1. Therefore, the slight differences observed in these two stress-strain curves areattributed to the Lode dependency term (Term 6). . . . . . . . . . . . . . . . . . . 131xviList of Figures6.4 The equivalent plastic strains at the onset of damage are predicted by Law 1 (a) andLaw 2 (b). For a detailed illustration, the θ¯-ε¯p0 curves are plotted under (c) η = -0.5,(d) η = 0, and (e) η = 1, respectively, together with the corresponding Îµ¯p-ÏÂ¯ profilesfor three cases (θ¯= -1, 0, and 0.88) at each η. . . . . . . . . . . . . . . . . . . . . . . 1356.5 Experimental setup in orthogonal cutting of single-track IN718 samples. . . . . . . . 1376.6 Laser confocal and SEM images of the single-track IN718 samples before and afterthe post-machining at different locations: (a) As-deposited surface; (b) Transitionregion between the as-deposited surface and the roughing surface; (c) Machinedsurface after roughing; (d) Transition region between the roughing surface and thefinishing surface; (e) Machined surface after finishing. . . . . . . . . . . . . . . . . . 1386.7 Residual stress reconstruction principles and results: (a) A schematic diagram to il-lustrate the principle of the stress reconstruction algorithm; (b) Reconstructed resid-ual stress distribution along the x direction; (c) Comparison of measured residualstress in the x direction and the reconstructed residual stress profile; (d) Comparisonof measured residual stress in the y direction and the reconstructed residual stressprofile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.8 Grain size information of DED-built single-track IN718 sample characterized byEBSD. (a), (b), and (c) refer to the grain boundary maps, IPF, and the distributionof fitted grain shape parameters (major diameters, minor diameters, and rotationangles) in xoy, xoz, and yoz planes, respectively. (d) illustrates how to calculate dg xand/or dg y based on the average ellipse grain shape parameters identified in eachplane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1436.9 Predicted residual stress distribution in the cutting velocity (x) direction under: (a)Law 1; (b) Law 2. Comparison between experimentally measured and numericallypredicted residual stress profiles: (c) Residual stresses in the x direction; (d) Residualstresses in the y direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1466.10 (a-c) Schematic diagrams illustrating the designed initial residual stress distributionwithin the workpiece featuring (a) compressive-dominant, (b) zero, and (c) tensile-dominant stress states in the chip formation region. The corresponding designedand reconstructed residual stress profiles at the selected path are compared, whichare referred to as (d) Initial profile 1, (e) Initial profile 2, and (f) Initial profile 3,respectively. The two expanded areas are used to balance the designed initial residualstress states. As a result, after completing the residual stress reconstruction, theresidual stress distribution in the middle area of the workpiece will remain at thedesigned values, while the residual stress states in the whole structure achieve bothequilibrium and compatibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148xviiList of Figures6.11 (a) Schematic representation of initial residual stress distributions, grain size val-ues, and material laws across 9 orthogonal cutting simulation cases for IN718. (b)Simulated tangential (Ft) and feed (Ff ) forces per unit width in each case. Thecolor scale in (a) provides a quantitative representation of the initial residual stressdistribution in MPa within the workpiece and is used consistently throughout thesubsequent figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1516.12 Workpiece stress distribution during orthogonal cutting of IN718 at t = 1.35 ms for(a) Case 1; (b) Case 2; (c) Case 3; (d) Case 4; (e) Case 5; (f) Case 6; (g) Case 7; (h)Case 8; (i) Case 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1526.13 Evolution of temperature (TEMP), equivalent plastic strain (PEEQ), stress alongthe cutting velocity (x) direction (Ïxx), and state of stress parameters (η and θ¯) at aselected point within the chip formation region during 9 post-machining simulations:Subplots (a), (d) and (e) represent the TEMP and PEEQ curves for Cases 1-3, Cases4-6, and Cases 7-9, respectively; Subplots (b), (e), and (h) depict the Ïxx profiles forCases 1-3, Cases 4-6, and Cases 7-9, respectively; Subplots (c), (f), and (i) show ηand θ¯ traces for Cases 1-3, Cases 4-6, and Cases 7-9, respectively. . . . . . . . . . . . 1536.14 Distribution of η and θ¯ within the workpiece during post-machining simulationsunder the given time instants: Subfigures (a)-(c) represent the variation of η in theworkpiece at Cases 4-6, respectively; Subfigures (d)-(f) depict the variation of θ¯ inthe workpiece at Cases 4-6, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 1546.15 Residual stress distribution along the cutting velocity (x) direction: Subfigures (a)-(c) show three maps of initial residual stress; Subfigures (d)-(l) present the corre-sponding final residual stress patterns after completing the 9 post-machining simu-lation cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1576.16 Residual stress profiles before and after post-machining simulations: Subfigure (a)presents the initial residual stress profile (Initial profile 1) and the correspondingfinal residual stress profiles under Cases 1, 4, and 7; Subfigure (b) plots the initialresidual stress profile (Initial profile 2) and the resulting final residual stress profilesunder Cases 2, 5, and 8; Subfigure (c) details the initial residual stress profile (Initialprofile 3) and final curves under Cases 3, 6, and 9. . . . . . . . . . . . . . . . . . . . 1586.17 Evolution of temperature (TEMP), stress along the cutting velocity (x) direction(Ïxx), state of stress parameters (η and θ¯), and plastic strain along the cuttingvelocity (x) direction (PExx) at a selected point on final surfaces in post-machiningsimulations: Subplots (a) and (d) represent TEMP and Ïxx curves for Cases 1-3 andCases 4-6, respectively; Subplots (b) and (e) depict the traces of η and θ¯ for Cases1-3 and Cases 4-6, respectively; Subplots (c) and (f) show the PExx profiles for Cases1-3 and Cases 4-6, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159xviiiList of Figures6.18 Distribution and evolution of contact forces on the cutting tool: Subfigure (a)presents the distribution of shear contact force at t = 1.35 ms under Case 4; Subfig-ures (b) to (j) plot the evolution of the sum of local normal and shear contact forcefor Cases 1 to 9, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160B.1 Comparison of longitudinal residual stress in thin-wall DED-built structures betweenFEM model [41], the prediction from model 1 (considering all three stress compo-nents), and the prediction from model 2 (considering only body forces). . . . . . . . 186B.2 Comparison of measured [41] distortion curves for a DED-built thin-wall structure,the prediction from model 1 (considering all three stress components) and the pre-diction from model 2 (considering only body forces). . . . . . . . . . . . . . . . . . . 186C.1 Initial geometry and boundary conditions of the proposed 2D FEM-based modellingprocedure to estimate the relaxed residual stresses. . . . . . . . . . . . . . . . . . . . 189C.2 Comparison of residual stress profiles before and after stress relaxation due to spec-imen removal by WEDM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189D.1 Initial geometry and boundary conditions of the proposed 2D FEM-based modellingprocedure to estimate the relaxed residual stress. . . . . . . . . . . . . . . . . . . . . 191D.2 Comparison of residual stress profiles before and after stress relaxation due to spec-imen removal by WEDM (Vc = 86 m/min, H = 0.12 mm, and wet cutting). . . . . . 191xixList of Symbolsα Thermal expansion coefficient [Kâ1]Î±¯ Thermal expansion coefficient of mixed materials [Kâ1]Î±n Clearance angle [°]αA Thermal expansion coefficient of material A [Kâ1]Î±B Thermal expansion coefficient of material B [Kâ1]Î² Stress softening coefficientγ Differences in yield strength between von Mises and Tresca criteriaγ0 Rake angle [°](γxx, γzz, γxz) Stress responses due to a unit load acting on surface pointsÎ Gru¨neisen ratioÎ0 Gru¨neisen coefficientδij Kronecker symbolâl Length of each cell [mm]ât Time increment [s]âA Area of cells facing the environment [mm2]âV Volume of each cell [mm3]ε0 Emissivityε¯0 Effective emissivity of mixed materialsËÂ¯Îµ0 Reference plastic strain rateεm Strain tensor after the mth unloading stepεp Plastic strain tensorε¯p Equivalent plastic strainËÂ¯Îµp Equivalent plastic strain rateε¯p0 Equivalent plastic strain at damage initiationxxList of Symbolsε¯p0,JC Equivalent plastic strain at damage initiation (JC damage model)ε¯p0,MMC Equivalent plastic strain at damage initiation (MMC damage model)εre Residual strain tensorεvol Measured nominal volumetric strainζ Segment numberη Stress triaxialityη0 Reference stress triaxialityηabs Heat source absorption coefficientηnor Nominal compressive volumetric strainθ Lode angleθ¯ Lode parameterθ1 Material gradient angle in vertical-dominant FGM walls [°]θ2 Material gradient angle in horizontal-dominant FGM walls [°]θs Stagnation angle [°]Î Total segments numberÎrotation Rotation angle of the fitted grain [°]Îsawâtooth Saw-tooth included angle [°]λ Linear energy density [JÂ·mmâ1]Î»avg Average thermal conductivity of Eulerian materialsλf Thermal conductivity of cutting fluidsλw Thermal conductivity of workpiece materialλwave Wavelength of surface profile wave [mm]µf Dynamic viscosity of cutting fluidsν Poissonâs ratioν¯ Poissonâs ratio of mixed materialsνcf Kinematic viscosity of cutting fluidsνA Poissonâs ratio of material AνB Poissonâs ratio of material Bξ Segment numberxxiList of Symbolsξthird The third deviatoric stress invariantÏ Density [Kg·mâ3]Ï0 Reference density [Kg·mâ3]ÏÂ¯ Density of mixed material [Kg·mâ3]ÏA Density of material A [Kg·mâ3]ÏB Density of material B [Kg·mâ3]Ï Stress tensor [MPa]Ï0 Stefan-Boltzmann constant [WÂ·mâ2·Kâ2](Ï1, Ï2, Ï3) Three principle stressesÏbody Thermal stress tensor induced by body forces [MPa]Ïflow Flow stress without considering damage [MPa]ÏÂ¯flow Softened flow stress[MPa]¯ËÏflow Flow stress after damaged-induced softening [MPa]Ïhydro Thermal stress tensor induced by hydrostatic stress [MPa]Ïm Stress tensor after the mth unloading step [MPa]Ïm Mean principal stress [MPa]Ïre Residual stress tensor [MPa]Ïre final Final residual stress tensor [MPa]Ïre in Initial residual stress tensor [MPa]ÏrexÎ¾ Longitudinal residual stress in the ξth segment [MPa]Ïsurf Thermal stress tensor induced by surface normal traction [MPa]Ïtherm Total thermal stress tensor [MPa]ÏÂ¯therm Thermal stress values in the von Mises equivalent form [MPa]ÏÂ¯JC JC flow stress without considering damage [MPa]ÏMJC MJC flow stress without considering damage [MPa]ÏT Elastic trial stress tensor [MPa]ÏY Yield strength of mixed materials [MPa]ÏYA Yield strength of material A [MPa]ÏYB Yield strength of material B [MPa]xxiiList of SymbolsÏ Cutting fluid traveling distance [mm]Ψ Tilted angles of samples in XRD measurements [°](âTâx ,âTây ) Thermal gradients [KÂ·mmâ1]a Strain softening coefficientAf Eulerion region filled with cutting fluidsAw Eulerion region filled with workpiece materialAJC Initial yield strength of materials (JC plasticity model) [MPa]BJC Strain hardening modulus (JC plasticity model) [MPa]c0 Sound velocity(c1, c2, nMMC) Material constants in MMC damage modelcη Hydrostatic pressure coefficient(csθ, caxθ , fθ) Material constants in Lode dependency termCJC Strain rate hardening coefficient (JC plasticity model)Cp Specific heat capacity [JÂ·Kgâ1·Kâ1]C¯p Specific heat capacity of mixed material [JÂ·Kgâ1·Kâ1]Câp Modified specific heat capacity [JÂ·Kgâ1·Kâ1]CpA Specific heat capacity of material A [JÂ·Kgâ1·Kâ1]CpB Specific heat capacity of material B [JÂ·Kgâ1·Kâ1]Cpf Specific heat capacity of cutting fluid [JÂ·Kgâ1·Kâ1]CNORMF Normal contact force per unit widthCSHEARF Shear contact force per unit widthdε Strain increment tensordλ Plastic correction coefficientdÏ Stress increment tensor [MPa]di (i = 1â 5) Five coefficients in the JC damage modeldg x Average grain size in the x direction [µm]dg y Average grain size in the y direction [µm]dl Heat source diameter [mm](D, P , R, S) Material constants in TANH termxxiiiList of Symbols(Dc, mLW ) Material softening parametersDtotal Total damage variableDOC Depth of cut [mm]epitch Pitch distance [µm]E Youngâs modulus [MPa]E¯ Youngâs modulus of mixed material [MPa]Em Internal energyEA Youngâs modulus of material A [MPa]EB Youngâs modulus of material B [MPa]EH Hugoniot energyf Yielding function/surfacefA Volume fraction of material AfB Volume fraction of material BFf Feed force [N]Fr Radial force [N]F surf Surface normal traction [MPa]Ft Tangential force [N]F bodyx Body force in the x direction [MPa·mmâ1]F bodyz Body force in the z direction [MPa·mmâ1]G Shear modulus [GPa](Gxxh, Gzzh, Gxzh) Plane strain Greenâs functions due to a horizontal unit load [MPa](Gxxv, Gzzv, Gxzv) Plane strain Greenâs functions due to a vertical unit load [MPa]h0 Segment thickness [mm]h1 Maximum chip thickness [µm]h2 Minimum chip thickness [µm]hchip Average chip thickness [µm]hcomb Combined heat convection coefficient [WÂ·mâ2·Kâ1]hconv Heat convection coefficient [WÂ·mâ2·Kâ1]h¯conv Effective heat convection coefficient of mixed materials [WÂ·mâ2·Kâ1]xxivList of SymbolsH Uncut chip thickness [mm]H0 Total height of the deposit-substrate system [mm]Hξ Distance from the center of the ξth segment to the neutral plane [mm]Hdep Height of deposits [mm]Hl Modified heat source penetration depth [mm]Hlay Deposition layer thickness [mm]Hsub Height of substrate [mm](i, j, l) IndicesI Volumetric heat source density [WÂ·mâ3]Ib Moment of inertia for the beam cross section [mm4]k Thermal conductivity [WÂ·mâ1·Kâ1]k¯ Thermal conductivity of mixed material [WÂ·mâ1·Kâ1]kf Thermal conductivity of cutting fluids [WÂ·mâ1·Kâ1]kn JC flow stress in the nth loading step [MPa]kA Thermal conductivity of material A [WÂ·mâ1·Kâ1]kB Thermal conductivity of material B [WÂ·mâ1·Kâ1]kHP Hall-Petch constant [MPa·µmâ1/2]KA Bulk modulus of material A [MPa]KB Bulk modulus of material B [MPa]L Length of the substrate/deposit [mm]Lf Latent heat of fusion [JÂ·Kgâ1]Lmajor Major diameter of the fitted grain [µm]Lminor Minor diameter of the fitted grain [µm]m Unloading stepmJC Thermal softening coefficient (JC plasticity model)M Total unloading stepsMx Total equivalent bending moment [N·mm]Mxξ Equivalent bending moment in the ξth segment [NÂ·mm]n Deposition layer number/Loading step numberxxvList of SymbolsnË Plastic correction vectornlay Total deposition layer numbernJC Strain hardening exponent (JC plasticity model)OPD Heat source penetration depth [mm]p Pressurepm Hydrostatic stresspH Hugoniot pressurePl Heat source power [W]Pr Prandtl numberq von Mises equivalent stressQconv Energy loss rate due to convection [J]Qconvârad Energy loss due to convection and radiation [J]Ql Heat source energy [J]Qrad Energy loss rate due to radiation [J]r The third invariant of deviatoric stressre Tool edge radius [mm]rl Heat source radius [mm]rn Tool nose radius [mm]Re Reynolds numbers Slope of Us versus UpS1 X-ray elastic constant 1S2 X-ray elastic constant 2S Deviatoric stress tensor [MPa]ST Trial deviatoric stress tensor [MPa]t Simulation time [s]tidle Idle time [s]T Temperature [K] or [°C]T0 Room temperature [K]T corner Temperature of corner cells [K]xxviList of SymbolsT edge Temperature at edge cells [K]T f Cell temperature before unloading [K]T interior Temperature at interior cells [K]Tm Melting temperature [K]Tmax Maximum temperature in the deposit-substrate system [K]Trex Stress/Strain relaxation temperature [K]TOL Predefied tolerance in the Newton-Raphson methodUp Particle velocityUs Shock velocityvl Heat source traverse speed [mm·minâ1]Vc Cutting velocity [mm·minâ1]Vf Cutting fluid velocity at nozzle [mm·minâ1]Vfw Maximum cutting fluid on the machined surface [mm·minâ1]w Width of cut [mm]w0 Width of the deposit-substrate system [mm]wmax Maximum vertical distortion of the thin-wall structure [mm]wA Weight ratio of material AwB Weight ratio of material Bw(x) Distortion curve of the thin-wall structure [mm](x, z) Coordinates [mm](xâ², zâ²) Reference coordinates [mm](x0, z0) Center of the heat source [mm]xxviiList of AbbreviationsAM Additive ManufacturingCEL Coupled Eulerian-LagrangianCr ChromiumCu CopperDED Directed Energy DepositionEBSD Electron Backscatter DiffractionEDS Energy-Dispersive X-ray SpectroscopeEVF Eulerian Volume FractionFDM Finite Difference MethodFe IronFEM Finite Element MethodFGM Functionally Graded MaterialFVM Finite Volume MethodHAGB High-Angle Grain BoundaryHEA High Entropy AlloyIPF Inverse Pole FigureJC Johnson and CookLAGB Low-Angle Grain BoundaryLW Li and WierzbickiMC Mohr and CoulombMJC Modified Johnson and CookMMC Modified Mohr and CoulombMn ManganeseMo MolybdenumxxviiiList of AbbreviationsNb NiobiumNi NickelNRMSE Normalized Root Mean Square ErrorPE Plastic strainPEEQ Equivalant plastic strainRRM Radial Return MethodSEM Scanning Electron MicroscopeTEMP TemperatureTi TitaniumWEDM Wire Electrical Discharge MachiningXRD X-ray DiffractionxxixAcknowledgementsIn the West Coast,Near N 49°,Rain drizzles outside,Once lingers for half a year.I stand by the window,Tracing the path of my past four years . . .I first need to express my deepest gratitude to Dr. Xiaoliang Jin, my supervisor during my timeat UBC. Your guidance on conducting research critically, your encouragement in presenting ideasclearly, and your feedback on expressing thoughts logically have shaped me into the researcher I amtoday. I also would like to acknowledge my co-supervisor Dr. Adam Clare. I enjoy every discussionwith you. I appreciate all the constructive comments you have provided on my work. I am sincerelygrateful for your suggestions on my future. You will never know how much it means to me to workwith you both.I am grateful to my supervisory committee members, Dr. Yusuf Altintas, Dr. Gary Schajer,and Dr. Chadwick Sinclair, for their invaluable comments during my literature review and researchproposal exams.I extend my gratitude to all my academic collaborators: Dr. Chris Bennett and Dr. AlistairSpeidel from the University of Nottingham, Dr. Serafettin Engin from Pratt & Whitney CanadaCorp., Dr. Raja Kountanya and Dr. Seung-Yub Lee from Pratt & Whitney, and Dr. Tahany El-Wardany from the Raytheon Technologies Research Center. Their valuable expertise and technicalinsights have significantly enriched my research.I want to thank my colleagues in the Advanced Manufacturing Processes Laboratory, all theengineers at the Manufacturing Automation Laboratory, all the technicians at the Center for High-Throughput Phenogenomics, lots of UBC faculty members and staff at the Student Machine Shops,and many others, for their essential assistance in my experiments.I am especially thankful to my dear friends, for their constant help in my life, and in particular,I would like to thank all physical, informational, and emotional support from Dr. Duan.Finally, I would like to thank my parents and my sister for their unconditional love. Thank youfor inspiring me with the courage to explore the world. Thank you for the daily messages filledwith warmth. Thank you for guiding me to become the person I am today.Overall, thank you all for making this journey unforgettable!xxxDedicationDedicated to:My Parents Qingyu Yang and Suqiu Zheng;My Sister Shenxia Yang.xxxiChapter 1Introduction1.1 Background and MotivationRemanufacturing is the process of restoring damaged components to an âas-good-as-newâ condition,allowing them to function optimally with an extended operational life [142]. Aero-engine turbineblades serve as a representative example of such components due to extreme operating conditions,including high temperature, high pressures, and foreign objective damage [122]. Recently, therehas been a growing interest in academia and industry regarding the utilization of directed energydeposition (DED) followed by machining for remanufacturing purposes [80, 169]. Fig. 1.1 illustrateshow to repair a damaged turbine blade (cracks near the tip portion of the blade) through DED +machining processes. Studies have reported that for a repair volume of 1.56 kg on a turbine bladeusing DED + machining, a minimum 45% carbon footprint improvement and a 36% decrease intotal energy usage were achieved compared to the entire blade replacement [151].Figure 1.1: Representative procedures to restore a damaged turbine blade through DED + ma-chining processes: (a) Original/reconstructed blade; (b) Damaged blade; (c) Fracture surface afterpre-machining; (d) Deposited area by DED; (e) Repaired area after machining.11.1. Background and MotivationResidual stresses refer to the âlocked-inâ stresses in materials and structures, independent ofthe presence of any external loads [156]. As discussed in [138], residual stresses are self-equilibratingwithin the whole volume of the material or structure: the local areas of tensile and compressiveresidual stresses sum to create zero force and moment resultants. According to the length scale overwhich they equilibrate, residual stresses can be classified as macroscale residual stresses (Type 1),which balance over the structure size; mesoscale residual stresses (Type 2, the focus of this study),which equilibrate over 3 to 10 times the grain size; and microscale residual stresses (Type 3), whichexist within individual grains [171]. Because of their self-equilibrating character, the presenceof residual stress may not be easily apparent, and so they may be ignored during engineeringdesign [138]. However, the manufacturing processes typically generate residual stresses, with thefollowing mechanisms: non-uniform plastic deformation, surface modification, and material phaseand/or density changes [138]. Depending on their magnitudes and distribution, the contributionof residual stresses can be either beneficial or detrimental to the functionality and performance ofmetallic structures. As shown in Fig. 1.2, improperly distributed residual stresses in manufacturedmetallic components may lead to severe deformation [110, 113], cracking [130], and delamination[36], ultimately reducing their durability and service life.Figure 1.2: Failures of metallic components caused by manufacturing-induced residual stresses:(a) Dimension inaccuracy [110]; (b) Cracking [130]; (c) Delamination [36]; (d) Deformation [110,113]; (e) Stress corrosion [71]. These figures are reprinted with permission from Springer Natureand Elsevier.21.1. Background and MotivationAs a type of metal additive manufacturing (AM) process, DED employs a focused energysource to melt the feeding materials, which are then deposited onto the substrate or previouslydeposited layers. Apart from fabricating and repairing single-material structures, DED also offersunique capabilities in producing functionally graded materials (FGMs) by melting feedstocks withcustomized material compositions layer by layer [10, 53, 179]. Regardless of whether powder-or wire-fed methods, DED involves high-gradient bulk residual stress, owing to the dynamic andcyclic heating and cooling procedures caused by the moving energy source. In DED-fabricatedstructures, bulk residual stress is a critical concern, as it influences the geometric distortion [110,123]. Oftentimes, the regions within DED structures dominated by high-gradient tensile bulkresidual stress are featured by elongation and increased distortion. To fully leverage the potentialof DED technology, it is essential to understand how residual stress and distortion evolve withinDED components, and to plan the optimum deposition strategies accordingly for achieving desiredgeometries.In addition to the presence of high-gradient bulk residual stress, the industrial application ofDED processes is also limited by their poor surface quality [127]. As a result, subsequent machiningprocesses are usually employed to satisfy the designed surface integrity and dimensional accuracyof DED-produced metallic structures. When machining metallic components, non-uniform plasticdeformation of workpiece material close to the bottom of the cutting tool edge leaves residual stresson the final surface. Apart from residual stress caused by phase transformation, the final conditionsof residual stress on machined surfaces rely on the accumulated thermo-mechanical loads exertedby the cutting tool. In addition, the final residual stress state on machined surfaces varies with theuse of cutting fluids [40], the generated chip morphologies [107], as well as the initial residual stressstates [154] and microstructural features [24], particularly in the machining of DED-fabricatedmetallic structures. Serving as one of the most fundamental factors in surface integrity, machinedsurface residual stress plays an important role in determining the fatigue life of finished components[99, 170]. For example, the presence of compressive residual stress on machined surfaces is beneficialfor anti-fatigue properties and could offset the impact of surface micro-defects, while high surfacetensile residual stress may lead to shorter fatigue life. Therefore, critical assessments of the finalresidual stress conditions on machined surfaces are essential.Due to the inherently âlocked-inâ nature of residual stresses, it is extremely challenging todirectly observe and evaluate them [138]. The existing methods to quantitatively determine thedistribution of residual stresses within metallic components can be divided into three categories:experimental measurements [138], analytical modelling [159], and numerical modelling [107]. Theexperimental measurements range from X-ray diffraction (XRD) [132] to hole drilling [116] and slit-ting [29], which can be categorized as non-destructive, semi-destructive, and destructive techniques.However, these measurement techniques are typically time-consuming and labor-intensive. On theother hand, analytical and numerical modelling methods provide an alternative for quantifying thedistribution of residual stresses. By incorporating physical laws of mechanics, heat transfer, andmicrostructure evolution, the predictive models enable an understanding of residual stress forma-31.2. Thesis Objectives and Outlinetion in both DED and machining processes, which are crucial in achieving process optimizationand ensuring the functional performance of metallic components.1.2 Thesis Objectives and OutlineThe main goal of this research is to develop physics-based models that are able to determine thegeneration of bulk residual stress distribution in DED-created structures and surface residual stressprofiles after machining, to inform the process strategies for achieving satisfactory performancesand functionalities of metallic components, particularly for remanufacturing purposes. The criticalquestions driving this research are:Question 1: How can DED-induced bulk residual stress be predicted efficiently?Question 2: How can machining-induced surface residual stress be accurately characterized?Question 3: How does DED-induced initial bulk residual stress affect final surface residualstress during machining?To answer these questions, this Ph.D. thesis comprises seven chapters, which are briefly de-scribed as follows.Chapter 1 details the background and significance of utilizing DED + machining technologiesfor the remanufacturing of metallic components, highlighting the necessity of modelling residualstresses in both DED and machining processes. It also explains the research objectives and outlinesthe thesis structure.Chapter 2 presents a new analytical and practical model that examines the thermo-mechanicalbehavior in single-material DED processes, reveals the DED-induced bulk residual stress formationmechanism, and quantifies the deposition parameters on the final bulk residual stress distributionin DED-fabricated single-material structures.Chapter 3 extends the single-material DED model in Chapter 2 to focus on the materialresponse in DED of FGM processes, investigates the effects of actual material composition ratioson bulk residual stress distribution in DED-produced FGM structures, and proposes novel DED ofFGM planning strategies to engineer the bulk residual stress distribution.Chapter 4 introduces an FEM-based numerical modelling procedure to simulate the tool,workpiece, and fluid interactions in wet machining processes, to effectively predict surface residualstress induced by wet machining and quantitatively evaluate the reduction in the surface tensileresidual stress due to the application of cutting fluids.Chapter 5 uses the FEM-based numerical modelling procedure developed in Chapter 4 toanalyze the coupling effects of shear localization in chips and micro-wavy features in machinedsurfaces on the fluctuation of surface residual stress along the cutting velocity direction in high-speed machining processes featuring serrated chip formation conditions.Chapter 6 further improves the FEM-based numerical modelling procedure developed inChapter 4 by integrating the initial residual stress states and microstructure features to eval-uate their cumulative interactions on generating the final residual stress on machined surfaces,41.2. Thesis Objectives and Outlinespecializing in the post-machining of DED processes.Chapter 7 summarizes the key conclusion of this study and highlights the major contributionsto the advancement of knowledge in the manufacturing field. In addition, it also discusses somepotential future research directions for modelling residual stresses in DED, machining, and post-treatments of AM structures.Overall, Chapter 2 and Chapter 3 are used to answer Question 1, while Chapter 4 andChapter 5 provides solutions for Question 2. Besides, the answer for Question 3 can be foundin Chapter 6. For clarity, Fig. 1.3 illustrates the graphical overview of this thesis.Figure 1.3: Graphical outline of this thesis.5Chapter 2Bulk Residual Stress inSingle-Material Directed EnergyDeposition Structures2.1 Introduction and Literature ReviewThe formation of DED-induced residual stress in single-material structures is described as follows:when a new layer is deposited, as shown in Fig. 2.1, the higher temperature gradient near the meltpool first generates compressive stress in the heating process due to the restriction of its thermalexpansion by the surrounding material, then it is replaced by tensile stress during the shrinkage ofthe melt pool material in the cooling process. In particular, the previously deposited layers restrictthe shrinkage of the new layer during cooling, leading to tensile stress in the latter. To maintainequilibrium, the tensile stress that develops during DED is counterbalanced by the accumulationof compressive stress during the re-melting and re-solidification of previously deposited layers. Inaddition, solid-state phase transformations also play an important role in the formation of DED-induced residual stress [55] due to the volumetric changes, which introduce additional strains thatrelax the stress [32].Bulk residual stress in single-material DED structures diminishes deposition quality and in-creases the costs of subsequent stress relief treatments. Post build heat treatments present sig-nificant limitations requiring tremendous energy input and often requiring precision features tobe added subsequently. Therefore, understanding how DED process conditions determine resid-ual stress profiles can guide process optimization and control, in order to achieve satisfactorydimensional accuracy, therefore improving the mechanical properties of the components. However,experimental characterizations of residual stresses, such as XRD and hole drilling, are expensive,time-consuming, and may not always provide sufficient information due to poor spatial resolu-tion. Since the magnitude and distribution of residual stresses are related to various depositionparameters, including heat source power [32], traverse speed [119], deposition pattern [85], materialproperties [38], deposit dimensions and structures [110], as well as substrate preheating conditions[31], additional understanding is often provided by modelling techniques.62.1. Introduction and Literature ReviewFigure 2.1: Temperature and stress distribution during laser-based powder-fed single-materialDED processes along (a) Path 1, (b) Path 2, (c) Path 3. x, y and z denote heat source traverse(longitudinal), transverse, and build (vertical) directions, respectively. Paths 1, 2, and 3 representintersection lines between the top surface of the substrate and planes 1, 2, and 3, respectively.Points oi (i = 1, 2, and 3) are the middle points of the three lines, and oiy i (i = 1, 2, and 3) refersto their positive directions.Physics-based numerical simulations are able to evaluate the effect of the process parameterson the deposition quality and mechanical properties of DED-fabricated single-material structures,thereby reducing the number of experiments required to examine each deposition condition. Amongvarious modelling methods, FEM-based models have been used to predict melt pool formation [91],cladding profiles [79], temperature distribution [48], powder velocity fields [61], and stress states72.1. Introduction and Literature Reviewin the deposited component [7] during single-material DED processes. Specifically, existing FEM-based models for predicting DED-induced residual stress fields in single-material structures canbe divided into two categories: thermo-mechanical and thermo-metallurgical-mechanical models.Table 2.1 lists some of the representative works using FEM for bulk residual stress prediction insingle-material DED processes and the main results.Table 2.1: Overview of FEM-based models used for predicting the bulk residual stress in single-material DED structures reported in the literature. The evaluation of their relative strengths andweaknesses is a key motivation for the present work.References Categories Software RemarksDing et al., 2011[41]Thermo-mechanicalanalysisABAQUS Simplifying transient DED depositionprocesses into a steady-state thermo-mechanical analysis to predict bulk resid-ual stress for large-size metallic compo-nentsVogel et al., 2013[164]Thermo-mechanicalanalysisCFD-ACE+ andIn-house codeSimulating DED processes from powderparticles delivery, laser-powder interac-tions, melt pool formation, and solidifica-tion to predict bulk residual stressHeigel et al.,2015 [62]Thermo-mechanicalanalysisCUBIC Modelling the effect of interlayer idle timeon bulk residual stress formation and evo-lution using a measurement-based heatconvection coefficientMukherjee et al.,2017 [123]Thermo-mechanicalanalysisIn-house code andABAQUSComparing bulk residual stress distri-bution in Ti6Al4V and IN718 thin-wallstructures under the same processing pa-rameters to estimate energy input andlayer thicknessWalker et al.,2019 [165]Thermo-mechanicalanalysisIn-house code andABAQUSMapping bulk residual stress in single-track IN718 deposits based on realis-tic deposition profiles considering powdercatchment efficiencyLu et al., 2021[109]Thermo-mechanicalanalysisCOMET Mitigating bulk residual stress in DED-built Ti6Al4V walls by optimizing sub-strate dimensions and structures82.1. Introduction and Literature ReviewReferences Categories Software RemarksFang et al., 2015[45]Thermo-metallurgical-mechanicalanalysisMSC Marc Studying the effect of martensitic trans-formation temperature on bulk residualstress formation in DED-built stainlesssteel componentsBailey et al.,2017 [18]Thermo-metallurgical-mechanicalanalysisIn-house code andABAQUSTaking tempering effects into considera-tion in the prediction of DED-inducedbulk residual stress with multi-track andmulti-layer capabilitiesDong et al., 2023[42]Thermo-metallurgical-mechanicalanalysisANSYS Considering temperature-dependent in-herent strains and phase-dependent prop-erties for predicting bulk residual stress inTi6Al4V thin-wall structuresIt has been reported that the computational time required for numerically predicting the bulkresidual stress in single-material DED processes varies from days to months, especially in thefabrication of large-size industrial-grade components [67]. To enhance computational efficiency,various techniques have been developed, including layer agglomeration/lumping [111], flash heating[166], and inherent strain methods [98]. Instead of adding the material step-by-step with themovement of heat sources, these models involve the deposition of materials segment-by-segmentto avoid interlayer deposition time. Although the proposed methods share acceptable predictionaccuracy while reducing the computation time, there are some concerns: (1) Heat sources used inthe calculation need to be carefully modified and calibrated based on the equivalent volumetric heatabsorbed by the material; (2) The way to deposit the material segment-by-segment reduces thermaldissipation area. As a result, the heat loss through convection and radiation is underestimated inthe thermal cycle prediction; (3) Uniformly applying heat sources on the newly deposited segmentsleads to prediction errors in both temperature and stress fields, as the material properties aretemperature-dependent; (4) Sensitivity studies are required to determine the critical length andheight of each segment.Analytical modelling serves as an effective method to understand the detailed thermo-mechanicalresponse of the deposited materials in the DED process, and has already fulfilled the demand forrapid prediction of residual stresses in machining and welding. However, few analytical models areavailable in the literature for predicting bulk residual stress in DED-built single-material struc-tures. Mirkoohi et al. [119] developed a physics-based analytical model to predict the bulk residualstress in IN718 parts fabricated by DED. The model was able to predict both the in-plane andout-of-plane residual stress profiles, and all the predictions were based on temperature-dependentmaterial properties. However, it is assumed that the final residual stress is only related to the ma-terial response when the last layer is deposited, regardless of how many deposition tracks and layers92.2. Single-Material DED Process Modelling and Validationare involved in the fabrication process. This assumption results in an underestimation of residualstress in the previous deposition layers, because the accumulated temperature and plastic strainare neglected when updating the yield stress of previously deposited materials. Besides, the simu-lation fails to consider the incremental buildup of the layers due to material addition but focuseson how the materials behave under the movement of the heat source. Compared to the materialaddition, the preactivated layer blocks the interactions between the heat source and the previouslydeposited material during deposition, resulting in an underestimation of the peak temperature inthe previously deposited layer. In addition, preactivating the entire layer at the beginning of thedeposition leads to expanded heat transfer zones during deposition, causing an overestimation ofthe thermal stress in the deposit-substrate system.An assessment of the literature shows a gap in providing computationally efficient and effectivemethods for the prediction of bulk residual stresses in single-material DED processes and theintegration with the process planning strategies for enhancing process performance. Specifically,the numerical models take extensive computational effort to simulate the bulk residual stresses forvarious process parameters, hindering process optimization and planning. In addition, the currentanalytical models do not consider the accumulated temperature and plastic strain on the yieldstrength and the material addition in the single-material DED process. To address these gaps, inthis chapter, we propose a new analytical model to predict the DED-induced temperature, bulkresidual stress, and geometric distortion with the consideration of material addition layer-by-layer.The detailed explanation and validation of the proposed model are presented in Section 2.2, followedby the analyses of the DED-induced stress evolution in Section 2.3. To guide the formulation ofDED strategies, in Section 2.4, two sets of DED parameters (including linear energy density andprocess planning parameters) are systematically investigated using the proposed model and theirinfluences on bulk residual stress distribution and geometric distortion in single-material structuresfabricated by DED are discussed.2.2 Single-Material DED Process Modelling and ValidationIn this section, the DED process for fabricating a single-material thin-wall structure is used todescribe the modelling procedures, while the developed model can be used generally for othergeometric features. Given the material properties, heat source parameters, and deposition processconditions, the model is executed iteratively to determine the temperature distribution (Section2.2.1) and residual stress states (Section 2.2.2) in the mid-longitudinal cross-section of a thin-wallstructure, as well as the vertical geometric distortion (Section 2.2.3). Specifically, as shown inFig. 2.2, for a point in the 2D space (x , z ) in the single-material DED deposition system and ata time instance t, the temperature at this point Tt(x, z) is updated according to the movement ofthe heat source, material deposition, and heat transfer. Then, the thermal stress Ïthermt (x , z ) isdetermined by the temperature and its gradients. To update the residual stress, a stress loadingprocess is carried out with the thermal stress as elastic trial stress and Johnson-Cook (JC) flow102.2. Single-Material DED Process Modelling and Validationstress for yielding criterion. The outcome of the loading process is defined as Ït(x , z ). Once thedeposition process is finished, the equilibrium conditions of the system are applied, where theremaining stresses represent the bulk residual stress Ïre(z ). Based on the obtained bulk residualstress, the distortion curve and the maximum vertical distortion of the structure, denoted as wmax,are calculated. The aforementioned modelling procedures enable a comprehensive understanding ofthe bulk residual stress formation in DED. This understanding is crucial as it directly contributesto the development and formulation of effective and efficient single-material DED strategies.Figure 2.2: An approach for modelling temperature, bulk residual stress and geometric distor-tion of DED-built single-material thin-wall structures. Sequentially coupled temperature, thermalstress and residual stress are calculated. The process maybe repeated as thermal âloadingâ and/orâunloadingâ takes place in subsequent layers of deposition. In the end, the obtained bulk residualstress profile is used to calculate the geometric distortion of the structure.The assumptions used in the developed modelling procedure are listed as follows: (1) Based112.2. Single-Material DED Process Modelling and Validationon the deposition conditions, the proposed model creates a 2D computational domain to featurethe material responses at the mid-longitudinal cross-section of a single-material thin-wall structure.In-plane phase transformation and any thermo-mechanical responses in the out-of-plane directionare not incorporated in the calculation. (2) At the end of the deposition, the 2D space will befilled by one material with homogeneous and isotropic properties. (3) Energy from the heat sourcegradually spreads through the material (in the conduction mode), causing localized melting andthen solidification in the vicinity of the heat source spot. Any complex thermal phenomena (likeconvection flows and the Marangoni effect) taking place in the melt pool and their effects on the heatloss are not included in the calculation. (4) During deposition, the thickness of each layer is assumedas constant while the top surface of the deposited layers is considered to be flat. Mass vaporizationand its effect on heat loss and the changes in the dimensions of each layer are not included. (5)The stress calculation relies on the plane strain condition. This assumption is applicable since thedeveloped model focuses on the material response on the mid-plane along the transverse directionof the fabricated structure. (6) In the prediction of geometric distortion, the thin-wall structure isassumed as a simply supported beam. The final distortion of the structure in the build (vertical)direction all comes from the bulk residual stress in the longitudinal direction (heat source traversedirection), and material heterogeneity is neglected.2.2.1 Prediction of temperature field2.2.1.1 Temperature modelling procedureThis section aims to predict the thermal cycles experienced by the deposit-substrate system insingle-material DED processes with the input parameters including the deposition dimensions (sub-strate/deposition length L, the height of the substrate Hsub, and the total height of the depositHdep), the heat source and deposition parameters (heat source power Pl, heat source traverse speedvl, heat source diameter dl, the height of each deposition layer Hlay, total deposition layer nlay, andidle time tidle), deposition patterns (unidirectional or bidirectional), as well as the temperature-dependent material properties (density Ï, thermal conductivity k, and specific heat capacity Cp).Given the above DED parameters, a 2D calculation domain is predefined and then discretized intosquare cells. As shown in Fig. 2.3(a), each cell shares an edge length of âl along both the horizontaldirection (heat source traverse direction) and vertical direction (build direction). A reference point(x, z ) is used to assign the center location of each cell in the Cartesian coordinate system XOZ.122.2. Single-Material DED Process Modelling and ValidationFigure 2.3: Schematics of temperature modelling in single-material DED processes: (a) 2D com-putation domain and boundary conditions at a deposition time of t ; (b) The finite difference method(FDM) used to update temperature.During the single-material DED process, the temperature varies at each cell, and the variationcan be captured based on elementary heat transfer laws. The detailed modelling procedure tocalculate the temperature variation at each cell center is explained hereafter. Specifically, at each132.2. Single-Material DED Process Modelling and Validationtime instant, the energy stored in the cells is the summation of the energy absorbed from the heatsource (Heating) and the heat flux by temperature gradients (Heat transfer). Please note that thedeveloped model not only tracks the temperature variation at cell centers during the deposition butalso extends to the corners of each cell (nodal temperature), which can be achieved by averagingthe center temperature values from neighboring cells based on the linear interpolation algorithm.i. HeatingTaking the deposition state in Fig. 2.3(a) as an example, with the movement of the heat source,the temperature of the cells exposed to the heat source is updated as [60]:Tt+ât(x, z) = Tt(x, z) +Ql(x, z)Ï(T ) Â·âV · Câp(T ), (2.1)where Tt(x, z ) is the transient temperature at the cell center (x, z ), âV is the volume of the cellwith a unit width, and Ï(T) is the temperature-dependent material density. C*p(T ) is the modifiedspecific heat capacity at T, which takes the effect of latent heat of fusion (Lf) on the temperaturedistribution into consideration, defined as:Câp(T ) =LfTm â T0 + Cp(T ), (2.2)where Cp(T ) is the temperature-dependent specific heat capacity, Tm is the melting temperature,and T0 is room temperature.In Eq. (2.1), Ql(x, z ) is the energy provided to the cell (x, z ) in a given time period by theheat source. Please note that the Ql(x, z ) varies with the type of heat source. Given a volumetricheat source [19] with the volumetric power density of I (x, z ) expressed as Eq. (2.4), Ql(x, z ) canbe written as:Ql(x, z) = ηabs Â·âV Â·ât · I(x, z), (2.3)I(x, z) =2PlOPD · Ïr2leâ 2(xâx0)2r2l(1â z â z0Hl), (2.4)where (x0, z0) denotes the center of the heat source, OPD is the optical penetration depth of theheat source under the conduction mode, Hl is the modified penetration depth based on OPD, rlis the radius of the heat source beam spot, ηabs is the heat source absorption coefficient by thematerial, and ât is the time increment during the calculation. The volumetric power density isuniformly distributed in each cell.ii. Heat transferAfter heating by the energy source, the temperature stored at each cell should then be updatedthrough the following heat transfer behaviors:142.2. Single-Material DED Process Modelling and Validation(i) Heat conduction from higher to lower temperature zones per time increment, and it canbe implemented with the 2D heat transfer governing equation expressed as [65]:k(T ) Â·(â2Tt(x, z)âx2+â2Tt(x, z)âz2)= Ï(T ) · Cp(T ) · âTt+ât(x, z)ât, (2.5)where k(T ) and Cp(T ) are the temperature-dependent thermal conductivity and specific heatcapacity of the material.(ii) Heat convection and radiation from the deposit-substrate system to the environment.Based on Newtonâs Cooling Law and Stefan-Boltzmann Law, the energy loss of the system throughthese behaviors at boundary cells in a given time ât can be written as:Qconv-rad(x, z) = âA Â·ât Â·{hcomb · [Tt(x, z)â T0] + ε0 · Ï0 Â·[T 4t (x, z)â T 40]}, (2.6)where âA is the area of the cell interfacing with the environment, ε0 is the emissivity, Ï0 isthe Stefan-Boltzmann constant (= 5.67Ã10-8 WÂ·m-2·K-4). hcomb is the combined heat convectioncoefficient, which can be written as:hcomb =(1hconv+âl2k(T ))â1, (2.7)where hconv is the heat convection coefficient.FDM is employed to solve the heat transfer equations mentioned above. While it is theoreticallypossible to model the temperature distribution in the deposit-substrate system using a moving heatsource, such an analytical approach typically requires assumptions that the entire deposition layer ispre-activated before heating, which overlooks the incremental material addition that characterizesDED. This assumption can lead to unrealistic heat transfer zones and overestimated thermal stressesin the deposition system. In contrast, FDM enables the incremental activation of elements whentheir temperature exceeds the melting point, thereby more accurately representing the depositionprocess. Additionally, FDM facilitates updating thermal boundary conditions such as convectionand radiation for edge and corner elements, which vary locally due to the incremental addition ofelements. This spatial and temporal evolution of temperature is difficult to achieve with closed-form analytical solutions, as they do not include the element addition. Nevertheless, using FDM,by calculating how many neighbors are surrounded by each element, it is efficient to update thethermal boundary conditions per time increment during the deposition process.As the prediction results are closely related to the cell size and time increment, their selectioncriteria are detailed in Appendix A. Therefore, under the given deposition conditions, the temper-ature field in the deposition and substrate areas can be tracked from the start of the deposition tothe end. For example, to update the temperature field in a given deposition time period, the cellsare divided into three groups (see Fig. 2.3(b)) based on the following heat transfer behaviors:Case 1: Interior cells. The change in the temperature in these cells is only related to heat152.2. Single-Material DED Process Modelling and Validationconduction. For a given time period, the cell temperature is updated as:T interiort+ât (x, z) =k(T )Ï(T )Cp(T )· ât(âl)2[Tt(x+âl, z) + Tt(xââl, z)+Tt(x, z +âl) + Tt(x, z ââl)]+[1â 4 · k(T )Ï(T )Cp(T )· ât(âl)2]Tt(x, z).(2.8)Case 2: Edge cells. Only one side of these cells is in contact with the environment. Apartfrom heat conduction, both heat convection and radiation need to be considered to update the celltemperature:T edget+ât(x, z) =k(T )Ï(T )Cp(T )· ât(âl)2{Tt(x+âl, z) + Tt(xââl, z) + Tt(x, z +âl)}+[1â 3 · k(T )Ï(T )Cp(T )· ât(âl)2]Tt(x, z)â âtâl(hcombÏ(T )Cp(T ))[Tt(x, z)â T0]â âtâl(Îµ0 · Ï0Ï(T )Cp(T ))[T 4t (x, z)â T 40].(2.9)Case 3: Corner cells. These cells have two sides adjacent to the environment. Similar to Case2, the temperature of these cells after a given time period should be updated based on the threeheat transfer behaviors, and they can be written as:T cornert+ât (x, z) =k(T )Ï(T )Cp(T )· ât(âl)2{Tt(x+âl, z) + Tt(x, z +âl)}+[1â 2 · k(T )Ï(T )Cp(T )· ât(âl)2]Tt(x, z)â 2 · âtâl(hcombÏ(T )Cp(T ))[Tt(x, z)â T0]â 2 · âtâl(Îµ0 · Ï0Ï(T )Cp(T ))[T 4t (x, z)â T 40].(2.10)For a large portion of cells, their heat transfer conditions do not remain unchanged throughoutthe deposition process. Considering the cell marked âXâ (see Fig. 2.4) as an example, it featuresthe heat transfer conditions listed in Case 3 at the deposition time of t1. As the deposition processcontinues, its temperature calculation procedure will transfer into Case 2 at t2 because of thenewly added cells, and then transfer to Case 1 at t3 till the end of the deposition due to thesequentially deposited layer. Besides, in FDM simulations, heat partitioning with the substrate ishandled through the thermal conductivity and the actual contact length between the deposit andthe substrate. When the laser moves across the newly added layer, heat is primarily conductedinto the substrate, especially during the early stages of deposition. Using FDM, the temperature of162.2. Single-Material DED Process Modelling and Validationeach node is calculated by accounting for the heat exchanged with its neighboring nodes, includingthose in the substrate. The amount of heat conducted into the substrate depends on the thermalproperties and the temperature gradient between the substrate and the newly added layer. Sincethe substrate acts as a large heat sink, a significant portion of the input energy is absorbed by thesubstrate, especially during the initial layer deposition.Figure 2.4: The change of boundary conditions at a cell of interest during a DED process.Boundary conditions are subject to change as the modes of heat transfer available to them alsochange.2.2.1.2 Validation of temperature predictionUnder the single-material DED process parameters listed in Table 2.2, the predicted thermal historyis compared with the experimental results and numerical calculation using the finite volume method(FVM) presented in previous studies [96, 123]. To validate the temperature model, the dimensionsof the melt pool, especially the depth, are generally more robust as they are less sensitive to localizedfluctuations. However, due to limitations in our facilities, we select the peak temperature valuesas an alternative. While peak temperature in each thermal cycles is more sensitive to factors likelaser diameter and may not fully capture melt pool dynamics, as suggested in [96, 123], it still canbe used for model validation.Table 2.2: DED process parameters for the validation of temperature [96, 123].Parameters ValuesHeat source power Pl [W] 2000Heat source traverse speed vl [mm·minâ1] 630172.2. Single-Material DED Process Modelling and ValidationParameters ValuesLayer thickness Hlay [mm] 0.9Substrate height Hsub [mm] 10Deposition length L [mm] 150Idle time tidle [s] 6Total deposition layer nlay 5Deposition pattern UnidirectionalFig. 2.5 shows the comparison among experimentally measured [96], numerically simulated[123], and our predicted thermal cycles during the deposition of a Ti6Al4V thin-wall by DED. Thisstructure is a vertical straight wall, deposited on a Ti6Al4V substrate. The purple dashed curverepresents the temperature evolution monitored by an embedded C-type thermocouple, which isattached on the substrate at the middle length of the top surface. Meanwhile, the green long-dashcurve shows the simulation results by a 3D transient heat transfer model using FVM. Finally, thered solid curve shows the prediction by the current model by FDM using the same parameters as theliterature [96], and temperature-dependent material properties of Ti6Al4V from Mukherjee et al.[123]. In our calculation, the Gaussian surface heat source model [165] is employed with an effectivebeam diameter dl of around 3.0 mm. The heat source absorption coefficient ηabs is assigned as 0.89,which is calculated based on the calorimetry experiments of Ti6Al4V conducted by Lia et al. [95].In their experiments, when the heat source power was set as 2000 W with the consideration ofpowder addition, the measured heat source power was 1780 W, which was determined by an OphirPhotonics Model 774258A power meter. The selections of cell size âl and the time increment âtmeet the criteria listed in Appendix A, which are set as 0.1 mm and 0.5 ms, respectively. It can beobserved that the temperature reaches the peak value once the heat source approaches the measuredpoint. Meanwhile, the maximum temperature in each thermal cycle decreases as the layer numberincreases, owing to the increase in the heat conduction distance between the heat-affected zone andthe measured point. As the heat source moves away, the temperature at the measured point firstdrops rapidly and subsequently decreases gradually. The decreasing rate is directly proportionalto the temperature gradient between the measurement point and the surrounding materials. Table2.3 compares the temperature prediction error for five peaks between the 3D FVM model and theproposed approach. Additionally, the mean absolute errors for each thermal cycle are listed in Table2.4. The results indicate that the simplified 2D heat transfer model based on FDM captures thetime-varying temperature field as precisely as the FVM model for a DED-built thin-wall structure.The deviation between the measured and our predicted temperature comes from the neglect of heattransfer in the out-of-plane direction. The results also suggest that, through a careful sensitivitystudy to determine appropriate element sizes and time step increments, the FDM can be used topredict the temperature in the selective laser melting process (with a laser diameter of around 50µm). Unlike the DED process, where incremental material addition with the heat source movement182.2. Single-Material DED Process Modelling and Validationis a key feature, this process involves a pre-filled powder bed across the entire layer before laserexposure. This condition aligns well with the assumptions underlying analytical moving heat sourcemodels described in [119]. Therefore, apart from FDM simulations, an analytical approach in [119]would also provide accurate temperature predictions.Figure 2.5: Comparison of the measured thermal cycles in [96] with respect to the predictions bythe FVM-based model in [123] and calculations by the current FDM-based model.Table 2.3: Peak temperature comparison and their prediction error. (Units: K)Peak No. 1 2 3 4 5Measurements in [96] 2325.4 1452.4 1300.9 1179.0 1086.7Predictions in [123] 2394.6 1556.2 1295.9 1109.8 1002.7(+3.0%) (+7.1%) (-0.4%) (-5.9%) (-7.7%)Current results 2100.9 1716.5 1454.1 1277.4 1156.7(-9.7%) (+18.2%) (+11.8%) (+8.3%) (+6.4%)Table 2.4: Mean absolute temperature error in each cycle compared to the measurements in [96].Cycle No. 1 2 3 4 5Predictions in [123] 8.0% 8.0% 5.9% 4.3% 5.8%Current results 8.9% 10.4% 10.0% 8.4% 7.5%192.2. Single-Material DED Process Modelling and Validation2.2.2 Determination of bulk residual stress2.2.2.1 Bulk residual stress modelling procedureThe magnitude and nature of bulk residual stress in the DED-fabricated single-material structuresare related to the thermal stress experienced by the material during the heating and cooling pro-cesses with the material addition. Determining the DED-induced bulk residual stress involves thefollowing three steps. Initially, the thermal stress is calculated based on the predicted tempera-ture and temperature gradients. Subsequently, the obtained thermal stress is imposed onto thedeposit-substrate system, layer by layer, with each layer corresponding to a thermo-mechanicalloading process. The loading process ends once all the deposition layers have been added to thesystem. Then, a thermo-mechanical unloading process is performed to release all elastic stresseswhile maintaining the plane strain condition. Finally, the remaining plastic stress represents thebulk residual stress states for the single-material structure.i. Thermal stress computationThermal stress in DED-built single-material components originates from the temperature gra-dients around the heat-affected zone caused by cooling following the rapid irradiation of the heatsource, repeated expansion and contraction of the deposited material because of the re-melting andre-solidification of the previously deposited layers, as well as the varying thermal expansion of thematerial at different temperatures due to the non-uniform heating. Given the temperature cal-culated in Section 2.2.1, thermal stress within DED-built single-material structures is determinedby three individual stress components, as shown in Fig. 2.6. For example, the thermal stressalong the x direction Ïthermxx (x, z ) is the sum of the stress components caused by the body forceÏbodyxx (x, z ), stress due to the normal traction acting on the top surface of the deposit Ïsurfxx (x, z ),and hydrostatic stress Ïhydro(x, z ). Appendix B discusses the necessity to include surface normaltractions and hydrostatic stresses in the thermal stress calculation. The calculation procedures foreach thermal stress component are explained below:(i) Ïbodyxx (x, z ) is the stress due to body forces Fbody acted at all other points in the deposit-substrate system. Specifically, Ïbodyxx (x, z, xâ², zâ²) is the reaction at the point (x, z ) due to the bodyforce acting on a point (xâ², zâ²), which can be expressed as follows:Ïbodyxx (x, z, xâ², zâ²) ={F bodyx (xâ², zâ²) Â·Gxxh(x, z, xâ², zâ²)+F bodyz (xâ², zâ²) Â·Gxxv(x, z, xâ², zâ²)}Â· (âl)2,(2.11)where Gxxh(x, z, xâ², zâ²) and Gxxv(x, z, xâ², zâ²) are the plane strain Greenâs functions [135], i.e., thestresses at (x, z ) resulting from a unit load at (xâ², zâ²), along the x and z directions, respectively.202.2. Single-Material DED Process Modelling and ValidationFigure 2.6: A schematic diagram of the thermal stress calculation at point (x, z ): (a) Body forces;(b) Surface normal traction; (c) Hydrostatic stress.Fbodyx (xâ², zâ²) and Fbodyz (xâ², zâ²) expressed in Eq. (2.12) represent the necessary forces in the x andz directions, respectively, which are used to maintain the system in its original shape due to the212.2. Single-Material DED Process Modelling and Validationtemperature change at point (xâ², zâ²).ï£®ï£°F bodyx (xâ², zâ²)F bodyz (xâ², zâ²)ï£¹ï£» =ï£®ï£°âÎ±(T )·E(T )1â2Î½(T ) · âTâx (xâ², zâ²)âÎ±(T )·E(T )1â2Î½(T ) · âTâz (xâ², zâ²)ï£¹ï£» . (2.12)In Eq. (2.12), α(T ), E(T ), and ν(T ) are temperature-dependent thermal expansion coefficients,Youngâs modulus, and Poissonâs ratio, respectively. âTâx andâTâz are the temperature gradients inthe x and z directions, respectively.Greenâs functions described in [135] are derived for a half-plane medium with homogeneousmaterial properties, which represents the response at one point in a homogeneous half-infinitedomain when a unit load is applied at another point within the half-infinite domain. However,in the DED process of single-material (Chapter 2) and FGM (Chapter 3) structures, temperaturevaries spatially due to the localized and cyclic heating and cooling caused by the movement of theheat source. Therefore, to predict residual stresses, it is necessary to consider the temperature-and composition- dependent material properties to calculate the actual forces applied at each pointin the deposit-substrate system during deposition and the responses of the system due to theapplication of a unit load. Nevertheless, the latter may conflict with the applicable conditionsof Greenâs functions presented in [135]. Considering that there will be no closed-form equationfor Greenâs function for inhomogeneous material, except we use methods for further modificationlike the perturbation method. Therefore, Greenâs function used in the thesis comes from thehomogeneous material condition as in its analytical form.To ensure the applicability of Greenâs functions illustrated in [135] for both Chapter 2 and Chap-ter 3, we further assumed that in the calculation of the responses caused by a unit point loading,the whole deposit-substrate system shares the same material properties as the average values at allthe material points in the system whose temperatures are below the stress relaxation point. Thisassumption can be justified because the material above the stress relaxation temperatures cannothold stresses. Therefore, there will be no thermal stress contribution involved for the residual stresscalculation. Meanwhile, the variations of Youngâs Modulus and Poissonâs ratio for single-materialand FGM (referred to as SS316 and IN718 in this thesis) structures caused by temperature and/orcomposition changes below the stress relaxation points are relatively slight, as discussed in [38] and[119].Based on the above assumptions, updating responses due to a unit load acting on the deposit-substrate system integrate the temperature- and composition- dependent material properties, andthe expressions can be simplified as functions of Poissonâs ratio and spatial coordinates, with thesame formulations as [135]. Then, to calculate thermal stresses, Greenâs functions expressed in[135] are applied in an incremental manner per time increment. Specifically, within a given timeincrement, Poissonâs ratio of the entire workpiece shares the same value as the focal element. In thenext time increment, to account for the effects of temperature and composition changes on the unitloading responses of the deposit-substrate system, Poissonâs ratio at the focal element is updated222.2. Single-Material DED Process Modelling and Validationaccording to the temperature and/or composition variation, and Poissonâs ratio of the whole systemis then adjusted based on the focal element. Consequently, Greenâs functions in [135] remain validat each time increment. A similar approach has been applied in a related paper [119].Following the above procedure, the values of G might be inaccurate due to the inappropriatePoissonâs ratio applied for the rest of the deposit-substrate system, particularly for the two regionswith significant differences in temperature/composition conditions. However, our calculations sug-gested that, when updating G, the relative positions between the points (x, z) and (xâ², zâ²) aredominant, while the change of G values due to the changes in Poissonâs ratio caused by the vari-ations in temperature (for both single-material and FGM structures) and composition (for FGMstructures) is insignificant, especially, considering that the materials used in FGM structure in thisthesis are SS316/IN718. It should also be noted that, the temperature- and material-dependentmechanical properties are considered in the force calculation. Instead of a unit force acting onthe point (xâ², zâ²), the actual force is a function of temperature gradients and material properties(herein, referred to as the thermal expansion coefficient, elastic modulus, and Poissonâs ratio),which dominants the thermal stress applied to the deposit-substrate system. Due to the use oftemperature- and composition-dependent thermal properties, the applied forces are more accuratethan the forces using constant material properties. Therefore, the calculation procedure of G isacceptable for the thermal stress calculation in this thesis.Besides, in our thermal stress calculations, we treat the entire deposit-substrate system as asemi-infinite domain and z = 0 is set on the top surface of each deposition layer. This assumptioncan be justified because, at each time increment, all elements in the deposit-substrate system(whether added or not) are initially assigned the same material properties corresponding to roomtemperature. Then, according to the actual material deposition, the stresses at each element willbe updated based on their temperature distribution. Specifically, for non-added elements, theyremain at room temperature, resulting in zero temperature gradients. Therefore, there will beno forces acting on the non-added elements to induce stress responses in the added elements.Meanwhile, to reflect physical reality, non-added elements are not supposed to bear stresses causedby temperature gradients on the added elements. This ensures that stress distributions remainlocalized to the deposited regions, while the stresses at the non-added region are zero per timeincrement.Last but not least, after deposition and cooling down, the top surface of the deposits is tractionfree in the build (z) direction as there is no external mechanical and thermal loading. To satisfythe force equilibrium condition, we assumed that thermal stresses in the z direction, which iscalculated by G and temperature gradients in the build (z) direction, are elastic stresses, andcannot be loaded into the deposit-substrate system. Therefore, in the calculation, the top surfaceof the deposits remains flat and no residual stresses are presented in the z direction of the structuresafter elastic stress unloading.(ii) Ïsurfxx (x, z ) arises from the normal traction Fsurf at surface points in the deposit-substratesystem. Let (xâ², zâ²) denote a point on the top surface of the current layer (zâ² = 0) or the previously232.2. Single-Material DED Process Modelling and Validationadded layer (zâ² = Hlay). The stress at (x, z) caused by the normal traction acting on a point (xâ²,zâ²) can be calculated as:Ïsurfxx (x, z, xâ², zâ²) = F surf(xâ², zâ²) · γxx(x, z, xâ², zâ²) Â·âl. (2.13)In Eq. (2.13), Fsurf(xâ², zâ²) represents the normal traction on (xâ², zâ²) with a temperature of T (xâ²,zâ²):F surf(xâ², zâ²) =α(T ) · E(T )1â 2ν(T ) · T (xâ², zâ²). (2.14)Besides, γxx(x, z, xâ², zâ²) is the stress response in the x direction at (x, z ) due to a unit loadacting on (xâ², zâ²), which can be calculated as:γxx(x, z, xâ², zâ²) =2zÏÂ· (xâ² â x)2[(xâ² â x)2 + z2]2 . (2.15)(iii) Ïhydro(x, z ) is equal to the stress in the x direction that is used to completely suppress thefree thermal expansion of the material at the point (x, z ), which can be expressed as:Ïhydro(x, z) = âÎ±(T ) · E(T ) · T (x, z)1â 2ν(T ) . (2.16)Based on the above analysis, the resulting thermal stress fields can be given in the integral form,as listed below:Ïthermxx (x, z) =â« Hsub+nHlay0â« L0[F bodyx (xâ², zâ²) Â·Gxxh(x, z, xâ², zâ²)+F bodyz (xâ², zâ²) Â·Gxxv(x, z, xâ², zâ²)]dxâ² dzâ²+â« L0F surf(xâ², zâ²) · γxx(x, z, xâ², zâ²) dxâ²+ Ïhydro(x, z),(2.17)Ïthermzz (x, z) =â« Hsub+nHlay0â« L0[F bodyx (xâ², zâ²) Â·Gzzh(x, z, xâ², zâ²)+F bodyz (xâ², zâ²) Â·Gzzv(x, z, xâ², zâ²)]dxâ² dzâ²+â« L0F surf(xâ², zâ²) · γzz(x, z, xâ², zâ²) dxâ²+ Ïhydro(x, z),(2.18)242.2. Single-Material DED Process Modelling and ValidationÏthermxz (x, z) =â« Hsub+nHlay0â« L0[F bodyx (xâ², zâ²) Â·Gxzh(x, z, xâ², zâ²)+F bodyz (xâ², zâ²) Â·Gxzv(x, z, xâ², zâ²)]dxâ² dzâ²+â« L0F surf(xâ², zâ²) · γxz(x, z, xâ², zâ²) dxâ²,(2.19)where n is the current (in progress) deposition layer number; Gzzh(x, z, xâ², zâ²), Gzzv(x, z, xâ², zâ²),Gxzh(x, z, xâ², zâ²), and Gxzv(x, z, xâ², zâ²) are the plane strain Greenâs functions [135]; Ïhydro(x, z )features the hydrostatic stress; γzz(x, z, xâ², zâ²) and γxz(x, z, xâ², zâ²) are the stress responses along thebuild and shear directions at the point (x, z ) because of a unit load applied on the surface points(xâ², zâ²), respectively, and they can be written as:γzz(x, z, xâ², zâ²) =2z3ÏÂ· 1[(xâ² â x)2 + z2]2 , (2.20)γxz(x, z, xâ², zâ²) =2z2ÏÂ· xâ² â x[(xâ² â x)2 + z2]2 . (2.21)Finally, considering the plane strain condition, the thermal stress in the transverse (y) directioncan be expressed as:Ïthermyy (x, z) = ν(T ) Â·[Ïthermxx (x, z) + Ïthermzz (x, z)]â α(T ) · E(T ) · T (x, z). (2.22)ii. Stress loadingDuring the stress loading process, plastic deformation occurs when the thermal stress reachesthe yield stress of the material, and the history-related stress-strain relationship during plasticdeformation needs to be considered. In this study, a radial return method (RRM) is employedto determine the strain and stress states based on the J2 flow plasticity theory. In this work,the yielding criterion we used is the JC plasticity law. This model combines strain hardening,strain rate hardening, and thermal softening effects on the yield surface. For accurately capturingcyclic loading effects, especially under multiple heating and cooling cycles, a kinematic hardeningmodel that updates the back stress tensor would provide a more realistic representation of materialbehavior. However, this work focuses on capturing the thermo-mechanical response in a limitednumber of cycles, where a nonlinear isotropic hardening law (herein, referred to as the JC plasticitylaw) offers a good prediction accuracy. In the future, for predicting large-sized structures withhundreds of thermal cycles, we plan to use more advanced plasticity models, such as a combinedisotropic-kinematic hardening law by modifying the original JC plasticity law, to better capturethe bulk residual stress.Taking the (n + 1)th loading process (in the fabrication of the (n + 1)th layer) as an example,the following three steps are required to update the strain and stress components:252.2. Single-Material DED Process Modelling and Validation(i) Elastic stress predictionThe thermal stress obtained in the previous section Ïthermn+1 serves as the elastic trial stress ÏTn+1.Then, the trial deviatoric stress tensor STn+1 is calculated as:STij,n+1 = ÏTij,n+1 â13ÏTll,n+1 · δij , (2.23)where δij is the Kronecker symbol.(ii) Determination of yielding criterionThe von Mises yielding criterion with the nonlinear isotropic hardening law is used to determinethe yield surface. Then, the yield function in the (n+1)th loading process can be expressed as [28]:fn+1 =12â¥STn+1â¥2 â13k2n, (2.24)where ||S||2 = Sjj2, kn is the yield stress of the material in the loading step n. In this work, thematerialâs yield stress is modelled by the JC plasticity law, expressed as [74]:kn = ÏÂ¯JC =(AJC +BJC · ε¯nJCp) · [1â CJC · ln ( ËÂ¯ÎµpËÂ¯Îµ0)]·[1â(T â T0Tm â T0)mJC], (2.25)where ε¯p, ËÂ¯Îµp, and ËÂ¯Îµ0 are the equivalent plastic strain, equivalent plastic strain rate, and referencestrain rate, respectively; AJC, BJC, CJC, nJC, and mJC are the five JC constants.If fn+1 < 0, the material does not yield and the loading process is elastic. Then, the elastic trialstress is the loading result (Sn+1 = STn+1) and the plastic strain remains unchanged, i.e., εp,n+1 =εp,n . Otherwise, the loading is elastoplastic, and the next step is required.(iii) Correction of plastic stressIn this step, the plastic correction direction nË and the correction coefficient dλ are determinedin order to correct the trial stress and to return the trial stress to the updated yield surface. nËrepresents the unit normal vector of the yield surface in the deviatoric stress space, which can beexpressed as:nË =STn+1â¥STn+1â¥. (2.26)dλ is obtained by solving the following equation [20]:g(dλ) = â¥STn+1â¥ â 2G · dÎ»ââ23· kn+1 = 0, (2.27)where G is the shear modulus.Since a nonlinear hardening constitutive behavior is used, the Newton-Raphson method isselected to determine the numerical solution of Eq. (2.27), and the method requires the followingsteps:(a) Variable initiation. The equivalent plastic strain ε¯0p,n+1, plastic correction coefficient dλ0,and the iteration number ζ are initialized as zero.262.2. Single-Material DED Process Modelling and Validation(b) Calculation of dλ. dλ can be solved based on the Newton-Raphson method [20]:ε¯ζ+1p,n+1 = ε¯ζp,n+1 +â23· dλζ , (2.28)g(dλζ) = â¥STn+1â¥ â 2G · dλζ ââ23·(kn +â23· dλζ · kâ²n), (2.29)gâ²(dÎ»Î¶) = â2Gâ 23· kâ²n, (2.30)dλζ+1 = dλζ â g(dÎ»Î¶)gâ²(dÎ»Î¶), (2.31)where ε¯ζp,n+1 and ε¯ζ+1p,n+1 are equivalent plastic strains in the ζth and (ζ + 1)th iteration step of the(n+1)th loading, respectively; dλζ and dλζ+1 are the correction coefficient in the ζth and (ζ +1)thiteration step, respectively; and kâ²n equalsdkndε¯p,n.(c) Convergence determination. If the absolute value of g(dλζ) is greater than the prede-fined tolerance (defined as 1Ã10â4, which is recommended in [20]) in the Newton-Raphson method,ζ = ζ +1 and the calculation should return to step (b).Once nË and dλ are determined, the equivalent plastic strain, equivalent plastic strain rate,plastic strain tensor and deviatoric stress tensor are updated as:ε¯p,n+1 = ε¯p,n +â23· dλ, (2.32)ËÂ¯Îµp,n+1 =Îµ¯p,n+1ât, (2.33)εp,n+1 = εp,n + dλ · nË, (2.34)Sn+1 = STn+1 Â·(â2kn+1â¥STn+1â¥). (2.35)Then, the stress components of the (n+ 1)th loading can be calculated as:Ïij,n+1 = Sij,n+1 +13ÏTll,n+1 · δij . (2.36)Following the above procedures, after each loading process, the strain and stress of each cell canbe determined. However, when the temperature at a material point exceeds the stress relaxationtemperature Trex, they begin to transform into a mushy zone, and cannot bear any strain andstress. In this work, when the temperature reaches Trex, plastic strain, equivalent plastic strain,and stress of the material points are reset as zero. For metallic materials, Trex is set to 80% of themelting temperature Tm [39].272.2. Single-Material DED Process Modelling and Validationiii. Stress unloadingAfter finishing the deposition process, the heat source is removed, and a stress unloading processis required to obtain the final residual stress profile in the deposited part. The boundary conditionsof the deposit-substrate system (the stress and strain conditions of the cells that need to be satisfied)are listed as follows [56]:{Ïrexx = f1(z), Ïreyy = f2(z), Ïrezz = Ïrexz = 0εrezz = f3(z), εrexz = f4(z), εrexx = εreyy = 0, (2.37)where εreij and Ïreij are the residual strain and stress components, respectively.Let εijf and Ïijf denote the strain and stress components at the end of the loading process. Tosatisfy the boundary conditions above, the non-zero stress and strain components, i.e., Ïzzf, Ïxzf,and εxxf, at each material point after the loading process, should be relaxed to zero step-by-step inthe unloading process, while the temperature Tf should be released to room temperature. Given Munloading steps, the increments of non-zero components in each unloading step are expressed as:dÏzz = âÏfzzM, dÏxz = âÏfxzM, dεxx = âÏµfxxM, dT = âTf â T0M, (2.38)where εxxf is determined by the following equation:εfxx =1â Î½2EÂ·(Ïfxx âÎ½1â ν · Ïfzz)+ (1 + ν) · α · T f + (εpxx)f , (2.39)where (εijp)f refers to the plastic strain in the x direction when the loading process is completed. Inaddition, based on the generalized Hookeâs Law and the plane strain condition, stress incrementsdÏxxf and dÏyyf are expressed as:dÏxx =E · dεxx + (1 + ν) · (ν · dÏzz â E · α · dT )1â ν2 , (2.40)dÏyy =ν · E · dεxx + (1 + ν) · (ν · dÏzz â E · α · dT )1â ν2 . (2.41)Analogous to the loading process, the above stress increments are taken as the elastic trialstress, and they are fed into the yield function to determine whether the unloading step is elastic orelastoplastic. If no plastic deformation occurs, the unloading process is purely elastic. Otherwise,RRM is employed to correct the released stress increments. Strain and stress increments remainunchanged as Eq.(2.38) in the following elastic unloading process. However, for the material thatundergoes the elastoplastic unloading process, once the plastic correction is completed, the strainand stress increments should also be adjusted accordingly. For example, if the mth unloading stepis elastoplastic, the strain and stress increments should be reset as Eq.(2.42) in the next unloading282.2. Single-Material DED Process Modelling and Validationstep.dÏzz = â ÏmzzM âm, dÏxz = âÏmxzM âm, dεxx = âÎµmxxM âm, (2.42)where εijm and Ïijm are the updated strain and stress components after finishing the mth unloadingstep, respectively. When the unloading steps are finished, the results Ïxx and Ïyy represent theresidual stress in the longitudinal (x) and transverse (y) directions, respectively.2.2.2.2 Validation of residual stress predictionThe residual stress predicted from the proposed model is compared with the results in [41] underthe deposition conditions listed in Table 2.5. Different from the temperature validation, this thin-wall structure is fabricated by a DED process defined as wire arc additive manufacturing. Thestructure has 4 layers, each with a deposition length and width of 500 mm and 5 mm, respectively.The material used is mild steel S355, and its temperature-dependent thermo-mechanical propertiesand JC parameters are listed in Table 2.6 and Table 2.7, and they are taken from Ding et al.[41]. In our calculation, the Goldak double ellipsoidal heat source model [41] is used, which hasa heat source absorption coefficient of 0.9. The front and rear semi-axis of the ellipsoid along theheat source traverse direction are 2 mm and 6 mm, respectively; and the semi-axis of the ellipsoidalong the build direction is 3 mm. The cell size âl and the time increment ât are assigned as0.5 mm and 4 ms, respectively. It should also be noted that the âaverageâ mechanical properties(Youngâs Modulus and Poissonâs ratio) are used in the calculation of Greenâs functions, while thetemperature-dependent properties are employed to update the temperature and the acting forcesto the deposit-substrate system in the thermal stress calculation.Table 2.5: Single-material DED process parameters for the validation of bulk residual stress [41].Parameters ValuesHeat source power Pl [W] 2246Heat source traverse speed vl [mm·minâ1] 500Layer thickness Hlay [mm] 2.0Substrate height Hsub [mm] 12Deposition length L [mm] 500Idle time tidle [s] 400Total deposition layer nlay 4Deposition pattern Unidirectional292.2. Single-Material DED Process Modelling and ValidationTable 2.6: Material properties of mild steel S315 used for the model presented here.Properties ValuesÏ [kg·mâ3] 7860k [WÂ·mâ1·Kâ1] 4.621Ã 101 + 0.047 · T â 1.117Ã 10â4 · T 2+4.841Ã 10â8 · T 3, T < 1573 [K];30, T â¥ 1523 [K].Cp [JÂ·kgâ1·Kâ1] 2.929Ã 102 + 0.749 · T â 4.966Ã 10â4 · T 2+1.084Ã 10â7 · T 3, T < 1573 [K];666, T â¥ 1523 [K].Lf [kJÂ·kgâ1] 270α [Kâ1] 1.11Ã 10â5 + 3.06Ã 10â9 · T, T < 1273 [K];1.50Ã 10â5, T â¥ 1273 [K].E [MPa] 2.239 Ã 105 â 8.204 Ã 101 · T + 0.070 · T 2, T <573 [K];4.292Ã 105â 4.000Ã 102 Â·T, 573 â¤ T < 973 [K];1.373Ã105â1.000Ã102 Â·T, 973 â¤ T < 1273 [K].1.000Ã 104, T â¥ 1273 [K]ν 0.259 + 8.099Ã 10â5 · TTrex [K] 1372Table 2.7: JC plasticity constants for mild steel S315 [41].AJC [MPa] BJC [MPa] nJC CJC mJC T0 [K] Tm [K] ËÂ¯Îµ0 [sâ1]448 782 0.562 0.0247 1.0 293 1723 0.001Fig. 2.7 compares the longitudinal residual stress predicted by a 3D steady-state thermo-mechanical FEM-based model [41] (shown as the red points) with the current model (representedas the blue solid line). The average prediction error in the single-material deposits between theproposed model and the result in [41] is around 12.6%. Both residual stress profiles initialize ascompressive at the substrateâs bottom surface and shift into tensile, with notable divergence in theirtransition paths at a distance of around 8 mm from the bottom surface of the substrate. In the caseof the residual stress profile predicted by the current model, the shift in the bulk residual stress fromcompressive (at the bottom surface of the substrate) to tensile (at the top surface of the substrate)occurs smoothly. Conversely, the residual stress profile predicted by the 3D FEM model begins to302.2. Single-Material DED Process Modelling and Validationshift from compressive to tensile at a distance of around 4 mm and concludes at about 8 mm. Thisprofile then levels off until it reaches the top surface of the substrate. These variations in the aboveresidual stress patterns can be attributed to the definition of the materialâs yield stress. Specifically,the current calculation employs the JC plasticity law, which takes into account the strain-hardeningeffect when determining the yield stress. During deposition, the region closer to the top surface ofthe substrate experiences more thermal loads and undergoes more plastic deformation than otherparts of the substrate. Consequently, plastic strain peaks on the top surface of the substrate andgradually decreases as the depth increases. The resulting plastic strain enhances the materialâsyield strength due to the strain hardening effect, leading to reduced tensile residual stress withinthe substrate and gradual transitions in stress states. In contrast, the material defined in the 3DFEMmodel maintains a constant yield stress after the plastic strain reaches or exceeds 0.1, failing toaccount for the strain-hardening effect. Therefore, the material yields at any location where plasticstrains are greater than or equal to 0.1, while their final residual stress equals the materialâs yieldstress. This leads to a constant tensile residual stress region near the top surface of the substrate(from 8 mm to 12 mm) and sharp shifts in stress states in the middle area of the substrate (from 4mm to 8 mm). In addition, the developed model shows notable benefits in computational efficiencyover the FEM-based simulations.Figure 2.7: Comparison of longitudinal residual stress in DED-built single-material thin-wallstructures between the FEM-based model [41] and the current model.Table 2.8 lists the computational time required for these two models. The results indicatethat by using the current model, the computational time required for the 3D FEM model can besignificantly decreased, resulting in a total time saving of around 98.4%. Furthermore, results fromthe current model suggest that bulk residual stress varied within the single-material deposits. Thevariation comes from the relaxation of stress and strain that, due to the remelting of previously312.2. Single-Material DED Process Modelling and Validationdeposited layers, occurs several times during the deposition of single-material multi-layer thin-wallstructures. This phenomenon of bulk residual stress variation has also been reported in [123].However, due to the limited resolution in the FEM-based analysis, the results in [41] show that theresidual stress in deposits is constant.Table 2.8: Comparison of computational time between the FEM-based model in [41] and thecurrent model.Temperature analy-sis timeStress analysis time Total analysis timeModel in [41] 10 min 14 h, 46 min 14 h, 56 minCurrent model 5.9 min 8.6 min 14.5 minTime saving 41.0% 99.1% 98.4%2.2.3 Calculation of geometric distortion2.2.3.1 Geometric distortion modelling procedureBulk residual stress is associated with geometric distortion, specifically for DED-built thin-wallstructures. Therefore, the optimal DED strategies should aim to reduce distortion as much aspossible with respect to the geometrical intent. This section describes an analytical modellingprocedure to predict the geometric distortion as a result of the DED-induced bulk residual stress.The procedure predicts both the deformation curves and the maximum distortion displacement ofDED-built single-material thin-wall structures in the build (vertical, z) direction.To predict the geometric distortion, the deposit-substrate system is simplified as a simply sup-ported beam. This beam has a uniform width as the deposit, which shares the same dimensions(length and height) and bulk residual stress distribution as the deposit-substrate system. There-fore, the total deformation of the system is equal to the beam deflection under the plane bendingcondition. There is an equivalent moment acting on each support of the beam. This moment is cal-culated based on the longitudinal residual stress distribution. As shown in Fig. 2.8, the depositionsystem is separated into Î segments in the z direction, and each segment has a uniform thicknessof h0.Let ÏrexÎ¾(Î¾ = 1, 2, . . . , Î) and Mxξ(ξ = 1, 2, . . . , Î) denote the average longitudinal residualstress and the equivalent bending moment in each segment, respectively. Then, to balance theDED-induced bulk residual stress, the total equivalent bending moment Mx is calculated as:Mx = w0 · h0ÎâÎ¾=1(ÏrexÎ¾ ·Hξ). (2.43)322.2. Single-Material DED Process Modelling and ValidationFigure 2.8: Calculation of the equivalent bending moment caused by DED-induced bulk residualstress.In Eq. (2.43), w0 is the width of the deposit-substrate system; Hξ defines the distance from thecenter of the ξth segment to the neutral plane. Under the condition that the neutral plane refersto the middle plane of the beam, Hξ is calculated as:Hξ =H0 â h02â (ξ â 1) · h0, (2.44)where H0 is the total height of the deposit-substrate system. The actual location of the neutralplane depends on the profile of bulk residual stress. Therefore, Hξ should be adjusted accordinglyin the calculation.Finally, the distortion curve w(x) and the maximum vertical distortion wmax for DED-builtsingle-material thin-wall structures can be calculated based on the basic beam equations:w(x) = âMx · x2E · Ib · (Lâ x), (2.45)wmax =Mx · L28E · Ib (at x =L2), (2.46)where Ib is the moment of inertia for the beam cross-section.2.2.3.2 Validation of geometric distortionThe bulk residual stress profile (shown in Fig. 2.7) is employed to calculate the geometric dis-tortion of the DED-built single-material thin-wall structure under the deposition conditions listedin Table 2.5. In our calculation, the structure is simplified as a simple rectangular-section beam332.2. Single-Material DED Process Modelling and Validationwith a length of 500 mm, a height of 20 mm, and a width of 5 mm. To calculate the total equiv-alent bending moment caused by DED-induced bulk residual stress, the beam is separated into 40segments and each segment has a thickness of 0.5 mm. The predicted distortion curve (the bluesolid line) is shown in Fig. 2.9. Meanwhile, the experimental data (shown as red points in Fig.2.9) were measured via a Romer Arm 3D laser scanner on the top surface of deposits by Ding etal. [41]. The results indicate that the predicted distortion curve agrees well with the experimen-tal measurements. The average prediction error is around 12.6 %, while the maximum predictionerror occurs at the two ends with the maximum prediction error of 17.7 % at the left end. Theobserved asymmetry in the measurement error at the left and right ends could originate from theunmelted feeding material accumulating when the heat source is suddenly turned off at the end ofa scan path, which occurs consistently on one side due to the unidirectional deposition strategyused in this work. The deviation between measured and predicted distortion curves arises fromboth measurements and predictions. Potential measurement errors include insufficient reflectivityof the surface being scanned by the laser scanner in the experiment. On the other hand, predictionerrors such as inappropriate evaluation of material properties (especially Youngâs modulus) afterdeposition, simplification of cross-sectional area in estimating the moment of inertia for the depo-sition system, and disregard of structural deformation caused by other residual stress componentsmay also contribute to the deviation.Figure 2.9: Comparison between measured [41] and our predicted distortion profiles for a DED-fabricated single-material thin-wall structure.342.3. Analyses on Stress Evolution in Single-Material DED Processes2.3 Analyses on Stress Evolution in Single-Material DEDProcessesThis section explains the detailed physical mechanism of the stress evolution that shapes the finalbulk residual stress state in DED-built single-material thin-wall structures. According to the tem-perature evolution history, the deposit-substrate system can be categorized into two zones (namednew-added and heated zones) at each deposition time instant. In the new-added zone, the materialis supposed to be deposited onto the substrate or previously deposited layers once the temperatureexceeds the melting point. On the other hand, the heated zone comprises materials that were pre-viously added to the system. Depending on whether their temperatures exceed Trex, the currentreheated zone can be divided into upper heated (T â¥ Trex) and lower heated (T < Trex) zones. Bycomparing the temperature/strain/stress history in each zone in the deposition of the first and thesubsequent layers, plastic stress formation and evolution in the DED parts and their differences arediscussed hereafter.Fig. 2.10 shows the temperature/strain/stress evolution at selected points during the depositionof the first layer. This single-material structure is fabricated by DED processes under the depositionparameters listed in Section 2.2.2 used for bulk residual stress validation. During fabrication, whenthe heat source approaches the mid-length of the deposit, three points illustrated in Fig. 2.10 arelocated on the new-added, upper heated, and lower heated zones, respectively. For comparison, thestress states (both thermal and loading stresses) at the three material points are expressed in thevon Mises equivalent form, while ε¯p denotes the equivalent plastic strain. With the movement of theheat source, the stress state of material points is set as zero or is updated by taking the minimumvalue between thermal and yield stresses. At the end of the deposition, the final stress state isa result of a multi-step loading process. Taking point 1 (shown in Fig. 2.10 (a)) as an example,to update the plastic stress in the newly-added zone, a three-stage loading process is required,known as zero-stress, plastic loading, and elastic loading processes. Specifically, when the materialpoint is melted and added to the deposition system, it is free of any stress and strain (Stage 1).As the heat source moves away, the temperature at this point decreases due to heat dissipationinto the lower-temperature surroundings through conduction, convection and radiation. Once thetemperature falls to the stress relaxation value (Trex), the material begins to bear stress and strain(Stage 2). At this time period, thermal stress is greater than the current yield strength, resulting inplastic deformation. To update the plastic stress, thermal stress is corrected based on the currentyielding surface, and the corrected value (equal to the current yield stress) is taken as the loadingresult. At the same time, ε¯p starts to rise from zero. As the temperature decreases further, thermalstress becomes lower than the current yield stress, indicating the end of the plastic loading stateand the beginning of the elastic loading state (Stage 3). During this period, the loading result isthe calculated thermal stress, and ε¯p remains constant until the end of the deposition. On the otherhand, updating the plastic stress in the current upper and lower reheating zones needs a loadingprocess consisting of five and three stages, respectively. The evolution of temperature/strain/stress352.3. Analyses on Stress Evolution in Single-Material DED Processesat points 2 and 3 (see Fig. 2.10 (b) and (c)) illustrates how the materials respond in the currentupper and lower heated zones, respectively. As shown in Fig. 2.10 (b) and (c), the stress at the twopoints starts with only elastic loading. During this stage, the loading stress is equal to the thermalstress, while ε¯p remains zero. As the heat source approaches, the temperature at this locationincreases, leading to an increase in thermal stress. When the thermal stress exceeds the currentyield stress, plastic deformation occurs, and the loading result becomes the corrected thermal stress.Meanwhile, ε¯p starts to increase from zero. As the heat source moves closer, the temperature atpoint 2 surpasses Trex while point 3 is still in the plastic loading stage. Therefore, both the stressand strain at point 2 are reset as zero, while ε¯p at point 3 continues to increase. As the heat sourcemoves away, point 2 leaves the stress-free state and starts to hold stress and strain as point 3.Similar to the new-added points, these two points end up being loaded only in the elastic range.It is worth noting that the stress updating process at the material points follows the samepattern as the first layer in the deposition of the subsequent layer shown in Fig. 2.11. However,the evolution of ε¯p in the current heated zones is different. As shown in Fig. 2.11 (b) and (c), theplastic strains at point 2 (in the upper heated zone) and point 3 (in the lower heated zone) areinitialized as a non-zero value due to the plastic deformation in the deposition of the previous layer.After passing the zero-stress stage, plastic strain at point 2 starts to increase from zero, which isdue to the resetting of ε¯p for the points at the solid-liquid interface (T below 80% Tm for the solidand above 80% Tm for the liquid), as shown in Fig. 2.11 (b). At the end of the deposition, thefinal ε¯p corresponds to the plastic deformation that occurs after the zero-stress stage. However, asillustrated in Fig. 2.11 (c), the final ε¯p in the current lower heated zone is greater than the initialvalue. The increase is due to the occurrence of plastic deformation at the material points as theheat source approaches. This increase in ε¯p results in the change of yield stress at the materialpoint, consequently altering the plastic stress remaining in the previous layer.362.3. Analyses on Stress Evolution in Single-Material DED ProcessesFigure 2.10: Evolution of temperature/stress/strain at selected points during the fabrication ofthe first layer for a single-material DED thin-wall structure. The deposition conditions are the sameas the residual stress validation in Section 2.2.2. When the heat source moves into the mid-lengthof the deposit, the three points are located at (a) the new-added zone, (b) the upper heated zone,and (c) the lower heated zone, respectively.372.3. Analyses on Stress Evolution in Single-Material DED ProcessesFigure 2.11: Evolution of temperature/stress/strain at (a) point 1, (b) point 2, and (c) point 3in the fabrication of the second layer for a DED-built single-material thin-wall structure.382.4. Guidance for Single-Material DED Strategies2.4 Guidance for Single-Material DED StrategiesChoosing appropriate DED strategies is crucial for achieving the desired quality of the fabricatedsingle-material structures. This section aims to investigate and demonstrate the immediate use ofthe simulation results based on the developed model, which is experimentally validated and exploredin previous sections. According to the proposed modelling procedures, two sets of DED parametersare selected to investigate their effects on bulk residual stress and geometric distortion in DED-built single-material thin-wall structures, including the linear energy density (λ) parameters (heatsource power, Pl, and traverse speed, vl) and the process planning parameters (idle time, tidle, anddeposition pattern). The remaining DED parameters are the same as the values listed in Table2.5. The findings provide insights in designing the DED strategies, which proves the feasibility ofenhancing structural quality while reducing processing time, resulting in cost savings and improvedproductivity.2.4.1 Linear energy density parametersIn the DED process, λ is used to represent the amount of energy delivered to the deposit-substratesystem per unit length along the direction of heat source traverse speed, which is defined as theratio between Pl and vl. In this section, a parametric study is conducted to quantify the combinedeffect of Pl and vl on the bulk residual stress and distortion of DED-built single-material thin-wall structures. In the parametric study, the ranges of Pl and vl are determined as follows. Theminimum Pl = 2000 W and the maximum vl = 550 mm/min are selected such that the top surfaceof the substrate is completely melted during the deposition of the first layer. Considering theactual working condition and the deposition efficiency, the maximum Pl is set as 2600 W while theminimum vl is assigned as 400 mm/min. Under the given deposition conditions, the effects of Pland vl on various indicators of bulk residual stress (including top surface residual stress, change ofthe average residual stress in deposits, and stress reverse position) as well as the maximum verticaldistortion of the fabricated single-material thin-wall structures are illustrated in Fig. 2.12. Fig.2.12(b) refers to the difference between the residual stress obtained from the selected parametersets and the residual stress calculated using the parameters in Section 2.2.2.It can be seen that changes in λ lead to variations in both bulk residual stress indicators andgeometric distortion of the fabricated structure. Table 2.9 lists the results of residual stress anddistortion for three representative deposition cases. The comparison results between case 1/case2 and case 3 suggest that the increase of λ results in higher tensile residual stress on the topsurface of deposits and greater vertical distortion of thin-wall structures fabricated by DED. Forexample, when λ increases from 252 J/mm (case 3) to 300 J/mm (case 1 and case 2), the top surfacetensile residual stress rises by 101.4% (case 1) and 13.8% (case 2), respectively. Additionally, themaximum vertical distortion increases by 49.3% (case 1) and 26.8% (case 2), respectively. In theDED process, the formation of tensile residual stress is caused by restricted thermal shrinkageof the newly deposited material after the heat source is removed. The magnitude of the tensile392.4. Guidance for Single-Material DED Strategiesstress is related to the homogeneity of temperature gradients during the cooling process, while thetotal tensile residual stress is positively correlated with the maximum distortion of the DED-builtthin-wall structure. On the other hand, a lower λ results in less heat accumulation on the newlydeposited surface and, therefore more uniform thermal contraction of the surface material. As aresult, lower magnitudes of tensile residual stress and slight geometric distortion are achieved.Figure 2.12: Effect of Pl and vl on (a) surface residual stress, (b) change of the average residualstress in deposits, (c) stress reverse position, and (d) maximum vertical distortion. (e) shows atypically deformed single-material thin-wall structure fabricated by DED.402.4. Guidance for Single-Material DED StrategiesTable 2.9: Comparison of bulk residual stress indicators and distortion under three depositioncases, illustrating the utility of this modelling approach for appraising and ultimately optimizationof deposition parameters.Case No. 1 2 3Heat source power Pl [W] 2500 2100 2100Heat source traverse speed vl [min/mm] 500 420 500Linear power density λ [J/mm] 300 300 252Top surface residual stress [MPa] 765.8 423.8 380.3Change of depositsâ average residual stress [MPa] 94.8 -12.0 -50.8Stress reverse position [mm] 3.46 3.04 4.42Maximum vertical distortion [mm] 3.21 2.73 2.15However, when the same λ value is applied, a deposition condition characterized by lower Pland vl results in reduced magnitudes of tensile residual stress on the top surface of deposits andless geometric distortion of thin-wall structures. For instance, cases 1 and 2 are characterized bythe same λ of 300 J/mm. As Pl and vl are decreased from 2500 W and 500 mm/min (in case 1)to 2100 W and 420 mm/min (in case 2), the top surface tensile residual stress reduces by 44.7%and the maximum vertical distortion decreases by 15.0%. Lowering both Pl and vl while keepingλ constant means that less energy is input into the deposition system. At the same time, theabove deposition condition (with a lower Pl and vl) facilitates heat dissipation from the depositedmaterial, thereby reducing heat accumulation within the deposition system. Therefore, a reductionin thermal gradients and thermal cycles can be observed during deposition, resulting in decreasedtensile residual stress and increased geometric accuracy in the fabricated structures.2.4.2 DED process planning parametersThe enhancement of process efficiency can be achieved by implementing various DED strategies,such as increasing heat source traverse speed and layer thickness, minimizing idle time, as well asmodifying deposition patterns. These DED strategies are designed to reduce the overall fabricationtime, improve the material deposition rate, and minimize the non-productive time. Nonetheless,it is still imperative to maintain the quality of the fabricated parts. In this section, the developedmodelling procedure is employed to investigate the effects of idle times (tidle = 200 s, 400 s, and 600s) and deposition patterns on bulk residual stress distribution and geometric distortion of DED-built single-material thin-wall structures. The selection of tidle meets the requirement of ensuringthe newly deposited materials have fully solidified before adding the next layer, while not coolingcompletely to room temperature. At the same time, two deposition patterns, namely unidirectionaland bidirectional, are incorporated into the analyses. Given the above deposition conditions, thesimulation results are presented in Fig. 2.13.412.4. Guidance for Single-Material DED StrategiesFigure 2.13: Effect of tidle and deposition pattern on (a) surface residual stress, (b) change of theaverage residual stress in deposits, (c) stress reverse position, and (d) maximum vertical distortion.It is shown that increasing tidle reduces both tensile residual stress within deposits and ge-ometric distortion of single-material thin-wall structures. For example, as shown in Fig. 2.13,when tidle increases from 200 s to 600 s, the top surface tensile residual stress reduces by 28.1%(unidirectional)/28.6% (bidirectional), while the maximum vertical distortion decreases by 33.7%(unidirectional)/32.1% (bidirectional). By increasing tidle in the DED process, there will be moretime between the depositions of successive layers. This extended cooling time allows for better heatdissipation for achieving thermal equilibrium with the surrounding environment, thus reducing thethermal gradients and the built-up thermal stress in the deposit-substrate system. As a result, boththe magnitudes of tensile residual stress and geometric distortion are reduced. However, excessivelylong idle time reduces productivity and efficiency, and may introduce negative consequences, suchas oxidation and contamination. Moreover, the prolonged idle time fails to yield significant im-provements in reducing residual stress and minimizing geometric distortion. This is exemplifiedin Fig. 2.13, where increasing tidle from 400 s to 600 s does not proportionally reduce the surfacetensile residual stress and the maximum vertical distortion, compared to the results by increasingtidle from 200 s to 400 s.In addition, when tidle is constant, changing deposition patterns influence both the bulk residualstress distribution and geometric distortion in DED-built single-material thin-wall structures. Asshown in Fig. 2.13, by modifying the deposition pattern from unidirectional to bidirectional, there422.5. Summaryis an observed decrease of 2.1% (tidle = 200 s)/2.2% (tidle = 400 s)/2.8% (tidle = 600 s) in the topsurface tensile residual stress and 12.8% (tidle = 200 s)/11.3% (tidle = 400 s)/10.7% (tidle = 600 s)in the maximum vertical distortion, respectively. In the DED process, the unidirectional depositionpattern involves depositing materials in a single direction, leading to localized high-temperaturezones and steep temperature gradients in the deposit-substrate system. On the other hand, thebidirectional deposition pattern, compared to the unidirectional one, is more efficient in fabricat-ing thin-wall structures since it refers to depositing materials in back-and-forth movements. Inaddition, this zigzag pattern facilitates a more consistent heat input and a more even distribu-tion of temperature gradients in the deposit-substrate system. Therefore, it is observed that thebidirectional deposition pattern yields lower tensile residual stress in deposits and reduced geo-metric distortion of the structure compared to thin-wall structures fabricated by the unidirectionaldeposition pattern through DED.2.5 SummaryIn this chapter, a new practical and analytical model is proposed to predict the thermo-mechanicalresponse in single-material DED processes, including temperature, bulk residual stress, and geo-metric distortion. The computational efficiency of this model enables the analyses and optimizationof the process parameters and strategy planning for DED, which can guide the process planningconsidering both efficiency and part quality. The main findings are summarized as follows:(1) The developed model has the novelty of featuring the addition of the material with themovement of the heat source, capturing the DED-induced temperature distribution, bulk residualstress states in the longitudinal direction, as well as stress-induced geometric distortion in thebuild direction. In the fabrication of a 4-layer thin-wall structure made by Ti6Al4V, the averageprediction error of bulk residual stress in the deposit is below 12.6%, while the prediction errorfor the maximum vertical distortion is within 17.7%. In addition, the model exhibits significantadvantages in computational efficiency compared to the FEM-based simulations. Under the samedeposition condition, the model reduces the total computational time needed for a 3D FEM-basedmodel by 98.4%.(2) Generation of the plastic stress/strain follows a multi-stage loading process in single-materialDED processes. During fabrication, the stress loading process can be summarized by separatingthe deposit-substrate system into new-added, upper and lower heated zones. In the first layerfabrication, the loading starts at the zero-stress stage for the newly-added points, followed by plasticand then elastic loading stages. However, the process begins as elastic in the current heated zones,followed by another 4-stage and 2-stage loading processes, respectively, depending on whether thetemperature exceeds Trex or not. In the deposition of subsequent layers, the stress loading sharesthe same pattern as the first layer, except for the evolution of plastic strain at the current heatedzones.(3) Two sets of DED parameters (linear energy density parameters, including heat source power432.5. Summaryand traverse speed, and process planning parameters, including idle time and deposition patterns)are investigated using the validated model. The findings suggest that linear energy density primarilydetermines the tensile residual stress in the deposits and the maximum vertical distortion. Higherlinear energy density results in increased tensile residual stress and severe geometric distortion.However, when the linear energy density remains constant, deposition conditions characterizedby lower heat source power and traverse speed lead to reduced tensile residual stress and lessvertical distortion. On the other hand, regardless of the deposition pattern (unidirectional orbidirectional), increasing idle time reduces both tensile residual stress and geometric distortion.However, excessively prolonged idle time fails to proportionally reduce tensile residual stress andminimize geometric distortion.44Chapter 3Bulk Residual Stress in FunctionallyGraded Directed Energy DepositionStructures3.1 Introduction and Literature ReviewFabrication of FGMs is a critical technique because they unlock design freedoms and enable newapplications in various industrial fields [58, 136]. Recently, FGMs fabricated by DED have gainedincreasing interest in aerospace, biomedical, and automotive industries, due to their ability totransition material properties between dissimilar components and customize material propertiesto enhance overall performance and functionality [47, 163]. For example, in the aerospace field,FGMs in turbine blades provide the required thermal resistance at elevated temperatures whilemaintaining enhanced toughness at lower temperatures [87]. Moreover, in biomedical applica-tions, functionally graded dental implants can be engineered to optimize mechanical propertiesand achieve the intended improvements in biocompatibility and osseointegration [117]. These ap-plications highlight the transformative potential of DED-built FGMs as an essential technique foradvancing next-generation metallic components where customized material properties are crucial.As a fusion-based process, FGMs fabricated by DED are characterized by spatially non-uniformtemperature and stress evolutions due to gradual transitions in composition within structures [124].These variations govern the bulk residual stress distribution and geometric distortion of fabricatedcomponents. Woo et al. [173] experimentally investigated the bulk residual stress in S45C/P21FGM fabricated by DED on an SS316L substrate with variations in chemical compositions ateach interlayer. The obtained results suggest that by adding intermediate layers, profiles of bulkresidual stress along the longitudinal direction, which plays a significant role in the crack initiationand propagation within the part, changed from a C-shape to a sinusoidal waveform. Therefore,the dependence of bulk residual stress distribution and part quality on the DED process conditionsmakes it necessary to model and understand the physical mechanism of how bulk residual stress isgenerated in FGM structures. This understanding is crucial for designing novel DED processes toachieve optimized FGM part quality.When modelling the bulk residual stress in the DED of FGM, Mukherjee et al. [124] conductedFEM-based simulations in the DED of S2.25Cr-1Mo/IN800H and Ti6Al4V/IN800H structures us-ing ABAQUS, considering the convective flow effect in the liquid metal. The results quantitatively453.1. Introduction and Literature Reviewhighlight a reduction in bulk residual stress and distortion of fabricated graded joints comparedto dissimilar joints. Li et al. [92] investigated bulk residual stress and part distortion in DED ofSS304L/SS316L/IN718/Cu structures through FEM-based simulations using ANSYS. The studyshows that high-temperature gradients induced by the mismatch in the thermal expansion coef-ficient among different materials lead to large displacements at the interface region and upwardbending of the fabricated structure. Zhang et al. [189] observed that adding D22 nickel alloyas interlayers in the DED of copper on H13 steel reduces the bulk residual stress magnitude by50% and achieves a crack-free structure through FEM-based simulations and experimental studies.Ghanavati et al. [54] used an FEM-based approach to study bulk residual stress and distortionin the DED of SS316L/IN718 thin-wall structures. The results demonstrate that sharp changesin plastic strain and stress at material interfaces can be effectively attenuated to mild changes byproperly designing the compositional gradients in the DED process. Shan et al. [141] developed anFEM-based multiscale framework using the inherent strain method together with the lumping layermethod for rapid prediction of bulk residual stress in DED-created austenitic/ferritic structures.Nevertheless, the inherent strains were calculated based on the properties of individual materials,and the variation of material composition at the interfaces between different material regions wasneglected. As a result, a significant deviation was observed between the predicted and measuredbulk residual stress profiles in these interface regions.The existing contributions in the literature focus on FEM-based predictions using commer-cial software to predict the bulk residual stress in DED-built FGM structures, assuming an idealmaterial composition transition path between two/multiple base materials in the build direction.Therefore, the variations in actual material composition at interfaces between different base mate-rials caused by the remelting of previous layers in DED are neglected. In addition, given the highprocess design flexibility in DED of FGM, including factors such as material composition varia-tions among deposition layers, heat source power, feeding rate of feedstocks, heat source traversespeed and pattern, it becomes challenging to systematically analyze and design the process plan-ning strategy to engineer bulk residual stress distributions, to minimize part distortion or potentialcracks. To address this gap, first, this chapter extends the physics-based model in Chapter 2 topredict the bulk residual stress in the DED-built FGM structures. The prediction results focuson the longitudinal residual stress, which is primarily responsible for potential crack propagationand warping [57, 180]. Then, to validate the enhanced model, SS316/IN718 thin-wall structureswere fabricated by powder-fed laser DED processes, and the measured residual stresses by XRDtechniques were compared with the predictions. In the predictions, both the designed materialcomposition transitioning path and the actual composition variation identified by the elementalcomposition weight ratio within the FGM structures are employed, and their differences in quanti-fying the effect of mixed materialsâ properties on the bulk residual stress distribution are analyzed.Finally, the validated model is used to investigate the influence of the material gradient directionon the bulk residual stress distribution pattern in FGM structures characterized by tilted materialtransition regions. The obtained results provide insight for optimizing DED process variables (in-463.2. Modelling of Bulk Residual Stress in DED-built FGM Structurescluding heat source power, traverse speed, idle time, and deposition patterns), while also enablingthe design of novel functionally graded DED strategies based on a thorough understanding of theprocess mechanisms.3.2 Modelling of Bulk Residual Stress in DED-built FGMStructures3.2.1 Modelling principles and assumptionsIn this section, an analytical model developed in Chapter 2 for modelling residual stress in DED-built single-material thin-wall structures is significantly extended and improved to focus on FGMcomponents. The principles of the enhanced model are presented in this section as a basis bywhich any DED process arrangements or material combinations can be examined. By integrat-ing the material composition transition strategies (transition direction and gradient levels), theenhanced model predicts the temperature distributions, elastic-plastic stress evolution, and bulkresidual stress status of DED-built FGM thin-wall structures. Although the focus of this chapteris to analyze the thermal-mechanical responses of FGM structures fabricated by joining two basematerials (Materials A and B) through a graded transition zone, other transition strategies, suchas the use of intermediate or filler materials, can also be integrated into this model. In addition,several assumptions are implemented in this model: (1) The designed transition of material A tomaterial B follows a discontinuously graded manner, having multiple transition layers in the in-plane direction; (2) Each transition layer corresponds to a predetermined/measured compositionof the two materials; (3) Material compositions are uniformly distributed within each depositionlayer and in the out-of-plane direction; (4) Effects of dilution due to the remelting of the substrateor previously added layers on materialsâ composition changes are neglected.3.2.2 Evaluating material properties in transitional regionsIn powder-fed DED processes, fabricating transitional regions of FGMs begins with mixing twobase metal powders in a predefined ratio. These blended powders are then delivered throughcoaxial nozzles onto either the substrate or previously deposited layers. As a result, the physicalproperties of the mixed materials in transitional regions are related to the two base materials.Herein, the two base materialsâ properties are considered to be temperature-dependent, featuringhomogeneous and isotropic characteristics. Therefore, by specifying the volume fraction of thetwo base materials within the transitional regions, the effective physical properties of the mixedmaterials (including density ÏÂ¯ , thermal conductivity k¯, specific heat capacity C¯p, Youngâs modulusE¯, Poissonâs ratio ν¯, thermal expansion coefficient α¯, and yield strength ÏÂ¯Y ) can be calculatedbased on the approximation of the rule of mixtures [92, 93, 124, 188, 190]:ÏÂ¯(T ) = fA · ÏA(T ) + fB · ÏB(T ), (3.1)473.2. Modelling of Bulk Residual Stress in DED-built FGM Structuresk¯(T ) = [kA(T )]fA · [kB(T )]fB , (3.2)C¯p(T ) = wA · CpA(T ) + wB · CpB(T ), (3.3)E¯(T ) = fA · EA(T ) + fB · EB(T ), (3.4)ν¯(T ) = fA · νA(T ) + fB · νB(T ), (3.5)α¯(T ) =fA · αA(T ) Â·KA(T ) + fB · αB(T ) Â·KB(T )fA Â·KA(T ) + fB Â·KB(T ) , (3.6)ÏÂ¯Y (T ) = fA · ÏYA(T ) + fB · ÏYA(T ), (3.7)where T is the temperature, fA and fB are the volume fractions of materials A and B in thetransition regions (fA + fB = 1), respectively. ÏA/B, kA/B, CpA/B, EA/B, νA/B, αA/B, and ÏYA/Bare the density, thermal conductivity, specific heat capacity, Youngâs modulus, Poissonâs ratio,thermal expansion coefficient, and yield strength for material A/B, respectively. wA and wB arethe weight ratios of each material (wA + wB = 1), respectively. KA and KB are the bulk moduli(a function of Youngâs modulus and Poissonâs ratio) for the materials A and B, respectively. Therelation between weight ratios (wA and wB) and volume fractions (fA and fB) is expressed asfollows:fA =1[ÏA(T )ÏB(T )· wB + wA] · wA, (3.8)fB =ÏA(T )ÏB(T )[ÏA(T )ÏB(T )· wB + wA] · wB, (3.9)3.2.3 Bulk residual stress modelling procedureFig. 3.1 illustrates the developed modelling procedures when fabricating an FGM thin-wall struc-ture by DED. The first step in obtaining the bulk residual stress profile is to calculate the temper-ature field based on the general heat transfer governing equation together with the Gaussian heatsource model, Newtonâs law of cooling, and Stefan-Boltzmann law of radiation, as listed below:ÏÂ¯(x, z, T ) · C¯p(x, z, T ) · âTât= k¯(x, z, T ) Â·[â2Tâx2+â2Tâz2], (3.10)I(x, z) =2Pl · Î·absÏr2l Â·OPDexp(â2[xâ (x0 + vlt)]2 + (z â z0)2r2l), (3.11)Qconv = hconv(T â T0), (3.12)Qrad = Îµ0Ï0(T4 â T 40 ), (3.13)where t is the processing time; x and z refer to the spatial coordinates of the element (based onthe Cartesian coordinate system XOZ shown in Fig. 3.1) in the longitudinal (X) and the build (Z)483.2. Modelling of Bulk Residual Stress in DED-built FGM Structuresdirections, respectively; vl is the heat source traverse speed; I is the volumetric heat source density;Qconv and Qrad are the heat loss rates due to convection and radiation, respectively; Pl is the heatsource power; ηabs is the absorptivity of the heat source by FGM; rl is the radius of the heat source;OPD is the heat source penetration depth (equal to the layer thickness Hlay, in this study); x0 andz0 are the initial coordinates of the heat source in the coordinate system XOZ shown in Fig. 3.1;hconv is the heat convection coefficient; ε0 is the emissivity; Ï0 is the Stefan-Boltzmann constant;and T0 is the room temperature.Figure 3.1: Modelling of temperature and bulk residual stress in DED-built FGM thin-wallstructure. The thermal stress loading process occurs at each time increment ât. Ï1 and Ï2 referto the principal stress components.Analogue to [180], the above equations are solved by FDM to capture the rapid tempera-ture changes in the FGM-substrate system. The temperature changes result from the interactionsbetween the heat source and the FGM, as well as the heat transfer behaviors caused by temper-ature differences within the system (leading to heat conduction) and between the system and itssurroundings (resulting in heat convection and radiation). The implementation of this methodrequires discretizing the computational domain according to deposition parameters and updatingheat transfer conditions based on the addition of FGM with the heat source movement. In thisstudy, the mid-longitudinal cross-section of the FGM thin-wall structure is separated into squarecells with a cell length of âl, while the cell temperature is updated per time increment ât. Theselection of âl depends on the heat source size and the layer thickness [180], as expressed in thefollowing equation:493.2. Modelling of Bulk Residual Stress in DED-built FGM Structuresâl â¤ min{2rl5,Hlay4}. (3.14)Based on âl and the temperature- and composition-dependent material properties, ât can bedetermined as Eq. (3.15) to ensure computational stability [180]:ât â¤ min{ÏÂ¯(x, z, T ) · C¯p(x, z, T )4 · k¯(x, z, T ) · (âl)2}. (3.15)Once âl and ât are finalized, the temperature prediction in the FGM-substrate system beginsby activating all substrate cells at an initial temperature of T0, while deactivating all FGM cells.To capture the temperature distribution in the FGM-substrate system per ât, three steps areessential: heating, material addition, and heat transfer. During the heating process, the locationof the heated cells and their corresponding energy absorbed from the heat source are computedbased on the heat source model expressed as Eq. (3.11). As the heat source moves, FGM cells aresequentially added into the FGM-deposit system once their temperature exceeds the melting pointTmelt. Subsequently, the activated cells in the FGM-substrate system are categorized into threegroupsâinterior cells, edge cells, and corner cellsâbased on the number of cell edges exposed tothe environment, to facilitate the heat transfer process. For instance, under a given temperaturedistribution at the processing time of t (Tt), after ât, the temperature updates for all interior cells(T interiort+ât ), edge cells (Tedget+ât), and corner cells (Tcornert+ât ) of the FGM-substrate system follow Eqs.(3.16) to (3.18). The above three processesâheating, material addition, and heat transferâwilliterate until the deposition concludes.T interiort+ât (x, z) =k¯(x, z, T )ÏÂ¯(x, z, T ) · C¯p(x, z, T ) Â·ât(âl)2·[Tt(x+âl, z) + Tt(xââl, z) + Tt(x, z +âl) + Tt(x, z ââl)]+[1â 4 · k¯(x, z, T )ÏÂ¯(x, z, T ) · C¯p(x, z, T ) Â·ât(âl)2]· Tt(x, z).(3.16)T edget+ât(x, z) =k¯(x, z, T )ÏÂ¯(x, z, T ) · C¯p(x, z, T ) Â·ât(âl)2·[Tt(x+âl, z) + Tt(xââl, z) + Tt(x, z +âl)]+[1â 3 · k¯(x, z, T )ÏÂ¯(x, z, T ) · C¯p(x, z, T ) Â·ât(âl)2]· Tt(x, z)â âtâl· hcombÏÂ¯(x, z, T ) · C¯p(x, z, T ) · [Tt(x, z)â T0]â âtâl· ε¯0(x, z) · Ï0ÏÂ¯(x, z, T ) · C¯p(x, z, T ) Â·[T 4t (x, z)â T 40].(3.17)503.2. Modelling of Bulk Residual Stress in DED-built FGM StructuresT cornert+ât (x, z) =k¯(x, z, T )ÏÂ¯(x, z, T ) · C¯p(x, z, T ) Â·ât(âl)2·[Tt(x+âl, z) + Tt(x, z +âl)]+[1â 2 · k¯(x, z, T )ÏÂ¯(x, z, T ) · C¯p(x, z, T ) Â·ât(âl)2]· Tt(x, z)â 2 · âtâl· hcombÏÂ¯(x, z, T ) · C¯p(x, z, T ) Â·[Tt(x, z)â T0]â 2 · âtâlÎµ¯0(x, z) · Ï0ÏÂ¯(x, z, T ) · C¯p(x, z, T ) Â·[T 4t (x, z)â T 40].(3.18)In Eqs. (3.17) and (3.18), ε¯0(x, z) refers to the effective emissivity of the FGM at the cell (x, z);and hcomb is the combined heat convection coefficient, which is expressed as:hcomb =(1h¯conv(x, z)+âl2k¯(x, z, T ))â1, (3.19)where h¯conv(x, z) denotes the effective heat convection coefficient at the cell (x, z). The calculationof ε¯0(x, z) and h¯conv(x, z) follows the approximation of the rule of mixture.According to the predicted temperature field in the FGM-substrate system, three steps arerequired to obtain the final bulk residual stress profile: thermal stress calculation, thermal stressloading, and elastic stress unloading. The flow chart to illustrate how to calculate the bulk resid-ual stress in DED-created FGM structures is shown in Fig. 3.1. As described in [180], thermalstress Ïtherm can be calculated by three stress components: stresses induced by body forces (causedby temperature gradients in the FGM-substrate system), stresses due to surface normal traction(caused by the temperature changes on the top surface of the FGM-substrate system), and hy-drostatic stress. In the calculation of thermal stress, updating Greenâs functions follows the sameprocedures as discussed in Section 2.2.2, which assumes that both Youngâs modulus and Poissonâsratio are temperature- and composition-independent, while calculating the acting forces to theFGM system is based on the temperature- and composition-dependent properties. During thermalstress loading, the determination of yielding in FGM relies on the von Mises yielding criterion withthe nonlinear isotropic hardening law. For instance, to determine the yield surface in the (n+1)thloading process (fn+1), the yield function is expressed as:fn+1 =12â¥STn+1â¥2 â13(ÏÂ¯Yn)2 , (3.20)where ÏÂ¯Yn is the effective yield stress in the loading step n, which is equal to the JC flow stress, asexpressed in Eq. (2.25); STn+1 refers to the trial deviatoric stress tensor in the loading step (n+1),which can be calculated as:STij,n+1 = Ïthermij,n+1 â13ÏTll,n+1 · δij , (3.21)where Ïthermij,n+1 denotes the thermal stress components in the loading step (n + 1); and δij is theKronecker symbol.Depending on whether the thermal loading is elastic or elastoplastic, after each loading step,513.3. DED of FGM Experimental Validationthe stress components remaining in each cell will be either the loaded thermal stress (for elasticdeformation) or the corrected plastic stress (for elastoplastic deformation). In this study, once theloading step is determined as elastoplastic, the RRM is used to determine the corrected plasticstress, as detailed in [180], which adjusts the loaded thermal stress to fit the current yield surface.Finally, after finishing the thermal stress loading steps and unloading all elastic stresses fromthe FGM-substrate system, the remaining stresses will be the bulk residual stress in the FGMstructure. Stress relaxation of FGM is also included in the calculation. When updating the stressstates at the material transitional layers, stress relaxation occurs once the temperature T is greaterthan the minimum stress relaxation temperature Trex (approximated to be 80% of the meltingpoint) of the two base materials. Therefore, when adding one deposition layer, a newly addedFGM cell will experience three stress states: zero-stress state (T â¥ Trex), elastoplastic loading state(T0 â¤ T < Trex, while ÏÂ¯therm â¥ ÏÂ¯Y ), elastic loading state (T0 â¤ T < Trex, while ÏÂ¯therm < ÏÂ¯Y ), whereÏÂ¯therm refers to the thermal stress of the FGM expressed in the von Mises equivalent form.3.3 DED of FGM Experimental ValidationIn this section, SS316/IN718 FGM thin-wall structures were fabricated by powder-fed laser DEDprocess, and residual stress along the longitudinal (x) direction was measured by the XRD tech-nique. Under the same deposition condition, the analytical model was employed to predict thebulk residual stress profile of the FGM structure. Two groups of simulations were conducted: onebased on the designed weight ratios of SS316 and IN718 in each deposition layer, and the otherperformed under the actual weight ratios of the two materials detected by the energy-dispersiveX-ray spectroscope (EDS). All of the above will be detailed in the following subsections.3.3.1 Fabrication conditions of FGM structuresA commercial DED machine (Optomec LENS CS-250, USA) was used to fabricate SS316/IN718FGM thin-wall structures. This DED machine is equipped with a 1 kW laser system, and the laserspot size was set as 1.33 mm. Feedstocks were gas-atomized SS316 and IN718 powders (PraxairSurface Technologies, USA). As illustrated in Fig. 3.2, their chemical compositions were quantifiedusing a Hitachi SU3500 scanning electron microscope (SEM) equipped with a Bruker XFlash 6-10EDS detector. The obtained results will be used as references to identify the pure SS316 and IN718deposition regions. The fabrication strategy of the FGM samples (a total of 30 deposition layers)is also shown in Fig. 3.2. These samples were built by first depositing 6 layers of 100 wt% SS316on an SS316 substrate plate, and then increasing the weight percentage of IN718 in the depositionmixture by 25 wt% every 6 layers while reducing that of the SS316 powder until adding 6 layers of100 wt% IN718. The dimensions of the SS316 substrate plate were 50.8 mm (length) Ã 12.6 mm(height) Ã 4.1 mm (width). Before the deposition, the substrate was grit blasted and degreased byethanol. The tool path for creating the FGM sample followed a linear single-track multi-layer back-and-forth pattern, having 2 seconds of idle time between each layer. Before deposition, the initial523.3. DED of FGM Experimental Validationdistance between the deposition head and the substrate was set to 9.8 mm. During fabrication,the two metal powders were first mixed according to predefined weight ratios and then conveyed tothe melt pool by argon jets with a 7 liter/min flow rate. The argon gas also acted as the shieldinggas at a 9 liter/min flow rate to prevent oxidation. Other deposition parameters are detailed inTable 3.1. After deposition, the dimensions of the fabricated FGM parts were measured as 50.8mm (length) Ã 10.2 mm (height) Ã 2.1mm (width).Figure 3.2: Experimental setup, deposition strategy, and metal powder information used in thefabrication of DED-built SS316/IN718 FGM thin-wall structures. In the Cartesian coordinatesystem O-XYZ, the X, Y , and Z axes are parallel to the longitudinal direction, transverse direction,and build direction, respectively.Table 3.1: DED process conditions for fabricating the FGM thin-wall structures.Parameters ValuesLaser power 280-350 W 1Powder feed rate 7.85 g/minLaser traverse speed 508 mm/min1The laser power in the first two layers was set as 350 W, while it was set as 280 W for the remaining layers.This power adjustment can avoid uneven deposited surfaces caused by excessive heat accumulation.533.3. DED of FGM Experimental ValidationParameters ValuesZ-axis increment 0.254 mm3.3.2 EDS tests and analysesIn the fabrication of FGM by DED, the remelting of previously deposited layers and the convectionflow of mixed materials within the melt pool (Marangoni effect) cause significant variations inelement composition at the interfacial layers between sections with different SS316-IN718 ratios. Asa result, the approximation of the rule of mixture based on the designed material ratios becomes lessaccurate in capturing the thermo-mechanical response of FGM. Therefore, accurately featuring theactual material composition distribution within the FGM structures becomes essential. To achieveit, in this study, the FGM structure was first sectioned by wire electrical discharge machining(WEDM) at the mid-transverse cross-section (parallel to the yoz plane, as shown in Fig. 3.3).The sectioned samples were then ground and polished with waterproof metallurgical papers (AlliedHigh Tech) on a manual grinder-polisher (EcoMet 30, Buehler). Next, the prepared samples wereplaced in a Hitachi SU3500 SEM to perform EDS testing. Finally, by comparing the measuredelement ratios to the nominal element ratios (derived from the measurements of pure SS316 andIN718 metal powders in Section 3.3.1), the actual weight ratios of the two materials along the build(z) direction of the FGM component can be quantified.Fig. 3.3 illustrates the EDS measurement strategies and results. Before the EDS measurement,the mid-transverse cross-section of FGM structures was divided into 30 segments along the build (z)direction. Each segment (separated by the yellow dot line) corresponds to one deposition layer, witha dimension of 0.34 mm along the build (z) direction (the thickness of each deposition layer). EDSarea tests were performed on both the substrate and each segment of the FGM within the regionsselected by an âorange boxâ to obtain the average element composition. The âorange boxâ labeledâArea 0â indicates the substrate area, while other boxes are named according to their respectivedeposition layer numbers. Each âorange boxâ shares the same dimensions, i.e., a height of 0.24 mmin the build (z) direction and a width of 0.3 mm in the transverse (y) direction. To ensure accuracy,three scans were conducted in each segment. Fig. 3.3 (a)-(f) list SEM images, energy spectra, andrepresentative element composition weight ratios for the selected regions at the substrate (Area 0)and the FGM at layer 3 (Area 3), layer 10 (Area 10), layer 15 (Area 15), layer 20 (Area 20), andlayer 28 (Area 28), respectively. As expected, there is a gradual transition in the most abundantelement from iron (Fe) to nickel (Ni), with the measurement locations shifting from SS316-dominantregions to IN718-dominant regions along the build (z) direction of the FGM structure. In addition,changes in weight ratios can also be observed along the build (z) direction for other representativeelements. Specifically, both niobium (Nb) and titanium (Ti) exhibit an increase from zero, whereasthe changes in chromium (Cr) and molybdenum (Mo) are less pronounced. Red solid lines presentedin Fig. 3.4 (a)-(f) quantify the changes in weight ratios of the elements Fe, Cr, Ni, Mo, Nb, andTi along the build (z) direction of the SS316/IN718 structure, respectively. The starting point of543.3. DED of FGM Experimental Validationthe profiles (the zero-distance location) in Fig. 3.4 is set as the interface between the substrate andthe FGM. The results indicate that as the distance from the FGM-substrate interface increases,the weight percentages of different elements vary significantly along the build (z) direction of theFGM. For example, Fe decreases from around 58.2 wt% to 17.7 wt%, while Cr exhibits a slightincrease with an amplitude of about 2.1 wt%. Conversely, Ni shows a steady increase from 11.3wt% to 44.6 wt%. Mo remains relatively stable, fluctuating slightly around 2.5 wt% to 2.8 wt%.Additionally, Nb and Ti both start at 0 wt% and increase progressively to 4.9 wt% and 0.7 wt%,respectively.The weight ratios of elements (Fe, Cr, Ni, Mo, Nb, and Ti) measured from pure SS316 and IN718metal powders are used as references to quantify the actual weight percentage of the two materi-als within the FGM structure. Nevertheless, there are certain variations in elemental compositiondetected by EDS for the two metal powders used in the fabrication of FGM structures. These vari-ations can be attributed to multiple factors, such as chemical inhomogeneities arising from powderproduction processes, environmental contamination during handling and storage, and measurementuncertainties influenced by powder morphologies. As depicted in Fig. 3.4, these elemental incon-sistencies in SS316 and IN718 metal powders are highlighted by semi-transparent yellow and blueerror bands, respectively. These error bands are centered by green long-dashed lines and magentashort-dashed lines, representing the average weight ratio of each element acquired from SS316 andIN718 metal powders, respectively. Given these error bands, the actual weight ratio profile of eachelement in the FGM structure (illustrated by the red solid line surrounded by grey error bands inFig. 3.4) can be subdivided into three segments. Specifically, the segment overlapping with thesemi-transparent yellow error band denotes the pure SS316 region, the segment intersecting withthe semi-transparent blue error band refers to the pure IN718 region, while the remaining segmentcorresponds to the region where the two materials are combined by DED.Considering the dominance of specific elements within the SS316/IN718 structure, the varia-tion amplitude along the build (z) direction, and their exclusivity among different materials, weuse weight ratio curves of four elements (Fe, Ni, Nb, and Ti) measured from the metal powdersas criteria for determining the weight ratio of SS316 and IN718 materials in each deposition layer.Specifically, Fe and Ni are concentrated in the SS316-dominant and IN718-dominant regions, re-spectively. Nb and Ti are exclusive to IN718 materials. Table 3.2 outlines the four criteria andtheir upper and/or lower boundaries. By applying these boundaries, the actual weight percentageof the two materials in each deposition layer can be determined based on the location of each layerwithin the three regions (pure-SS316, pure-IN718, and combined regions) mentioned above. Sincethe total weight percentage of the two materials equals 1, herein examples are provided to calcu-late the weight percentage of the IN718 material in each deposition layer, as follows: Depositionlayers in the pure SS316 region contain 0 wt% IN718 material, whereas those in the pure IN718region have 100 wt% IN718 material. Meanwhile, at the combined region of the two materials,the determination of the actual weight ratio of the IN718 material depends on the monotonicityof the weight ratio curves at the selected element. When the elementâs weight ratio curve in the553.3. DED of FGM Experimental Validationcombined region increases monotonically from the SS316-dominant region to the IN718-dominantregion (See Fig. 3.4 (c), (e), and (f)), the actual weight ratio of IN718 is equal to the relativegrowth rate of the elementâs weight ratio in this layer compared to the weight ratio of the sameelement at the upper boundary of the SS316 powder region (denoted as the semi-transparent yellowerror bands, the same as follows). Conversely, when the elementâs weight ratio decreases steadilyfrom the SS316-dominant region to the IN718-dominant region (See Fig. 3.4 (a)), the actual weightratio of IN718 refers to the decline rate of the elementâs weight ratio in this layer, relative to theweight ratio of the same element at the lower boundary of the SS316 powder region.Table 3.2: Four criteria and their corresponding upper/lower boundaries to determine actualweight percentages of SS316 and IN718 materials in each deposition layer. (Units: wt%)No. Elements Pure SS316 layers Pure IN718 layers Combined layers1 Fe â¥ 58.0 â¤ 18.3 (18.3, 58.0)2 Ni â¤ 11.8 â¥ 42.7 (11.8, 42.7)3 Nb = 0 â¥ 3.8 (0, 3.8)4 Ti = 0 â¥ 0.6 (0, 0.6)Based on the identified weight ratios of SS316 and IN718 materials in each deposition layer,the material transition paths within the FGM structure are plotted in Fig. 3.5. Fig. 3.5 (a)presents the designed weight ratio transition path of SS316 and IN718 materials within the FGMstructure, while Fig. 3.5 (b)-(d) show the actual transition paths of the two materials identifiedby the four criteria: Criterion 1 (Fe), Criterion 2 (Ni), Criterion 3 (Nb), and Criterion 4 (Ti),respectively. As the number of deposition layers increases, all subfigures depict a compositionaltransition pathway from purely SS316 to entirely IN718. However, the characteristics of transitionpaths, including the shape and slopes of the curves, vary among the five subfigures. For instance,the designed transition path exhibits sharp changes in the composition ratios of the two materials,particularly at the interfaces between layers 6-7, 12-13, 18-19, and 24-25, respectively. Conversely,the identified compositional transition paths across all deposition layers are smoother but haveunique intersection points (the points where the two curves intersect with each other) and variedinflection points (the points where significant changes in the slope of the curves occur). As aresult, depending on the selected criterion, the proportion of SS316-dominant or IN718-dominantregions within the FGM structure will be different. Material properties, such as density, specificheat capacity, and yield strength, used for the bulk residual stress calculation are closely relatedto the weight ratio distribution of the two materials within the FGM structure. Therefore, it isessential to quantify how the deviation between the designed and measured weight ratios of the twomaterials within the FGM structures affects the material response in the DED process. In the nextsubsection, both the designed and identified weight ratio transition paths of the two materials arefed into the analytical model to feature their differences in the prediction of bulk residual stress.563.3. DED of FGM Experimental ValidationFigure 3.3: SEM images, energy spectra, and measured element weight ratios at the transversecross-section of DED-produced FGM thin-wall structures for the selected areas at (a) the substrate,(b) layer 3, (c) layer 10, (d) layer 15, (e) layer 20, and (f) layer 28.573.3. DED of FGM Experimental ValidationFigure 3.4: Measured weight ratio profiles of the elements (a) Fe, (b) Cr, (c) Ni, (d) Mo, (e) Nb,and (f) Ti at the transverse cross-section of FGM thin-wall structures along the build direction.The profiles start at the interface between the substrate and the FGM. The measured elementweight ratios of SS316 and IN718 metal powders with certain error bands are plotted to identifythe mixed region of the two materials.583.3. DED of FGM Experimental ValidationFigure 3.5: Weight ratios of SS316 and IN718 materials at each deposition layer of the FGMstructure: designed weight ratios (a) and actual weight ratios identified by the four criteria basedon the measured weight percentage of the elements (b) Fe, (c) Ni, (d) Nb, and (e) Ti.593.3. DED of FGM Experimental Validation3.3.3 Residual stress measurements and predictionsLongitudinal residual stress (denoted as Ïrexx) at the mid-length of the FGM sampleâs longitudinalcross-section (parallel to the xoz plane shown in Fig. 3.6) was measured using an X-ray diffrac-tometer by Lambda Technologies Group. As illustrated in Fig. 3.6, to facilitate the X-ray beamreaching the measuring surface, the FGM sample was firstly sectioned by WEDM, and then elec-tropolished to remove the WEDM-affected layer. The residual stress measurements were performedusing a two-angle sine-squared-psi technique, employing the diffraction of manganese K-alpha ra-diation from the (311) planes of FCC structures for both SS316-dominant and IN718-dominantareas. The X-ray irradiated area was set as rectangular, with a length of 2.5 mm in the longitu-dinal (x) direction and a width of 0.5 mm in the build (z) direction. During measurement, thesample underwent angular oscillation within a range of ±1.5â¦ around the mean psi angles (rangingfrom 10â¦ to 50â¦). The aim was to broaden the integration of diffracted intensity across multiplegrains, thereby mitigating the effect caused by variations in grain size. Fig. 6 (d) depicts the distri-bution of residual stress measurement points (a total of 19 points) on the mid-longitudinal plane ofthe FGM structure. Three XRD measurements were conducted at each location described in Fig.3.6(d). After measurements, the values of the X-ray elastic constant (1/2S2) used for calculatingthe residual stress from the strain measured normal to the (311) planes were set as 7.1±0.13Ã10â6MPaâ1 in the SS316-dominant region (from point 1 to point 11) and 5.97± 0.03Ã 10â6 MPaâ1 inthe IN718-dominant region (from point 12 to point 19), respectively.Figure 3.6: Residual stress measurements of the FGM structure: (a) WEDM strategies; (b)Electropolishing; (c) Measurement area; (d) Detailed locations of each measurement area.603.3. DED of FGM Experimental ValidationMechanical and thermal properties of the FGM are temperature- and composition-dependentin the prediction. As contoured in Fig. 3.7, these properties were calculated based on the ma-terial database of SS316 and IN718 provided in [92]. On the one hand, the temperature depen-dence of the FGMâs thermo-mechanical properties is analyzed as follows. No matter the weightpercentages of these two materials, an increase in temperature leads to higher thermal conduc-tivity, specific heat capacity, and thermal expansion coefficient of the FGM, while density andYoungâs modulus decrease. However, changes in Poissonâs ratio are influenced by the dominantmaterial within the FGM. Specifically, Poissonâs ratio follows an increasing-then-decreasing trendin the SS316-dominant region and a consistently increasing trend in the IN718-dominant region,respectively. On the other hand, the composition dependence is highlighted below. Within the pre-scribed temperature ranges, the SS316-dominant region is characterized by a higher density thanthe IN718-dominant region. As the weight percentage of IN718 increases, there is an increase inthermal conductivity, specific heat capacity, thermal expansion coefficient, and Youngâs modulus.Furthermore, Poissonâs ratio demonstrates moderate variation with compositional changes when thetemperature is below 1073 K. However, when the temperature exceeds 1073 K, the IN718-dominantregion exhibits a higher Poissonâs ratio than the SS316-dominant region.Apart from the material properties mentioned above, the JC flow stress is used to determinethe FGMâs yield stress. The JC parameters of SS316 are taken from [161], and those for IN718 aresourced from [158]. Based on the JC flow stresses of SS316 and IN718, the approximation of therule of mixture is used to determine the effective yield stress of FGM in the material compositiontransitional regions. As described in Eq. (3.7), to obtain the JC flow stress in the materialtransitional region, the rule of mixture is applied to linearly interpolate the JC flow stress betweenSS316 and IN718 according to the compositional gradient. Other parameters used in the predictionare listed below. The coefficient of heat source absorptivity (ηabs) was set as 0.28 for SS316 [115]and 0.30 for IN718 [165], respectively. The heat convection coefficient (hconv) and the emissivity(ε0) of the two materials were taken from [92]. It should also be noted that, due to the analyticalframework of the model, the simulations of a 30-layer DED-built FGM structure with the depositionconditions mentioned in Section 3.3.1 take about 54 minutes (around 10 minutes for temperatureprediction and about 44 minutes for stress prediction) on an Intel (R) Core (TM) i9-1900K CPU@ 3.50 GHz, with 64 GB of RAM, demonstrating a higher computational efficiency compared totraditional FEM-based models.613.3. DED of FGM Experimental ValidationFigure 3.7: Temperature- and composition-dependent material properties used in the bulk residualstress prediction in the fabrication of SS316/IN718 thin-wall structures by DED: (a) Density; (b)Thermal conductivity; (c) Specific heat capacity; (d) Thermal expansion coefficient; (e) Youngâsmodulus; (f) Poissonâs ratio.623.3. DED of FGM Experimental ValidationFig. 3.8 compares the measured longitudinal residual stress (Ïrexx) at the selected locations(shown as red points connected by red trendlines) with the predicted residual stress profiles bythe developed model. Measurement results presented in Fig. 3.8 were corrected by removing therelaxed residual stress caused by material removal through WEDM + electropolishing. The relaxedstress values were estimated by a 3D FEM-based numerical modelling procedure, as presented inAppendix C. The FEM results indicate that 15.9% of the original DED-induced bulk residualstresses were relaxed due to the material removal by WEDM and electropolishing. Consideringthe necessity to perform WEDM + electropolishing before XRD measurements, we have expandedthe original measurement error bars to account for the bulk residual stress variation caused bythese processes. The predicted bulk residual stress profile in Fig. 3.8(a) was obtained using thedesigned material composition transition path. In contrast, Fig. 3.8(c)-(e) present the simulatedbulk residual stress profiles based on the actual material composition transition paths identified bythe four criteria: the weight percentages of elements Fe, Ni, Nb, and Ti, respectively. The resultssuggest that both the experimental measurements and predicted bulk residual stress profiles exhibita compressive residual stress state close to the substrate. This compressive residual stress shiftsto a tensile state near the mid-location of the FGM structure and continues to increase, withfluctuations, up to the top surface of the FGM structure. However, a significant deviation can beseen between the measured and predicted bulk residual stress profiles near the top region of theFGM structure, specifically between the measurements at points 13 and 14 and the simulated valuesat a distance of 7 mm from the substrate. The discrepancies could be attributed to uncertaintiesin residual stress measurements through XRD and simplifications in bulk residual stress modelling.First, the measurement error may come from the unmatched X-ray elastic constants, which wereapplied to compute the residual stress based on the measured strains. Specifically, the X-ray elasticconstants, calibrated using stress-free IN718 powders, were employed to determine the residualstress in the IN718-dominant region (from points 12 to 19). However, points 13 and 14 werelocated in the middle area of the deposition layer 20 and the interface between deposition layers21 and 22, respectively. According to EDS results, at least 20 wt% of SS316 powders were mixedwith IN718 powders in these regions. Consequently, applying the same X-ray constants in theseareas may introduce certain errors in the measured residual stress values. Second, the predictionerrors could arise from neglecting variations in the materialâs mixing ratios within each depositionlayer. In the simulation, it was assumed that each deposition layer shared consistent weight ratiosof SS316 and IN718. These ratios were either derived from the designed material transitional pathor calculated based on EDS measurements by quantifying the weight percentages of representativeelements. However, the actual weight percentages of the two materials were closely related tothe amount of the two metal powders being consistently delivered by the argon gas and beingeffectively melted by the laser. The EDS results shown in Fig. 3.4 also suggest that even forsingle material deposition layers, material composition variation still can be observed within eachdeposition layer, due to uncertainties in powder delivery and melting processes during fabrication.Therefore, the assumption of uniform weight ratios of SS316 and IN718 across each deposition layer633.3. DED of FGM Experimental Validationin the longitudinal direction may have contributed to the observed prediction errors.In addition, differences can also be observed among the five predicted bulk residual stressprofiles. Specifically, using the designed weight ratios of SS316 and IN718 in each deposition layer,the simulated bulk residual stress profile follows an abrupt, step-like transition from the compressiveresidual stress near the substrate to the tensile state at the top surface of the FGM. However, eachâstepâ is not entirely flat; residual stress peaks at the beginning of each âstepâ and then graduallydecreases. This contrasts with smoother, more gradual transitions observed in the predicted bulkresidual stress profiles using the actual weight percentages of the two materials in each depositionlayer. Based on the designed weight ratios, the abrupt changes in the simulated bulk residual stressprofile are caused by the sudden changes in material composition ratios. These abrupt changesin bulk residual stress occur at the interface regions between the two mixed materials, and thenumber of âstepsâ corresponds to the number of designed material composition regions within theFGM structure. As the designed material composition region transitions from SS316 to IN718,the corresponding yield stress at each region increases, leading to an increase in residual stress. Inaddition, peaks in residual stress at the start of each material transitional region (at the beginning ofeach âstepâ) can be attributed to the plastic strain accumulation in the deposition of the followinglayers, and the residual stress gradually decreases in each transitional region can be explained bythe stress relaxation caused by remelting in the subsequent layer deposition.The deviations between measurements and simulated bulk residual stress profiles are quantifiedby normalized root mean square error (NRMSE). Using the designed weight percentages of SS316and IN718 materials in each deposition layer, the NRMSE value is calculated as 22.59%. Comparedto the âdesignedâ conditions, the predictions based on the actual weight percentages of the twomaterials identified by the weight ratios of elements Fe, Ni, and Ti share lower NRMSE values of21.55%, 21.56%, and 21.21%, respectively, indicating a closer fit to the experimental measurements.It is also noticed that the maximum variation of the NRMSE values based on the above threeelements is about 1.65%, which demonstrates the consistency of the predictions using the threeindividual elements. However, the highest NRMSE value (around 26.42%) is obtained under Nb-based predictions, which is due to its smaller proportion in the material compared to other elements,making the results more influenced by measurement noise. Therefore, when EDS tests are available,it is recommended to use the actual weight percentage in each deposition layer identified by theelemental weight ratios of Fe, Ni, or Ti for predicting the residual stress distribution, as theyprovide more accurate and reliable predictions in thermo-mechanical responses of FGM comparedto the designed ratios and Nb-based prediction. Nonetheless, considering the accessibility andeffectiveness of performing EDS, predictions based on the designed ratios can also be acceptablefor industrial applications where such high accuracy is less critical.643.3. DED of FGM Experimental ValidationFigure 3.8: Comparison between measured residual stress points and the simulated bulk residualstress profiles based on (a) designed weight percentages of SS316 and IN718 in each depositionlayer; and actual weight percentages of the two materials identified by the weight ratios of elements(b) Fe, (c) Ni, (c) Nb, and (d) Ti.653.4. Bulk Residual Stress in DED-built FGM Structures Featuring Adapted Transitional Regions3.4 Bulk Residual Stress in DED-built FGM StructuresFeaturing Adapted Transitional RegionsIt is feasible to fabricate FGM samples characterized by adapted material transitional regionswith material gradient flexibility in DED [53, 118, 150]. This can be achieved practically byadjusting material mixing ratios along the longitudinal (x) direction and the build (z) directionsimultaneously. Consequently, alterations will occur in the distribution patterns of bulk residualstress within the part. Based on the experimentally validated model, this section aims to quantifythe effect of material gradient angle on the bulk residual stress distributions, which can inform thedevelopment of novel DED strategies with various material gradient strategies. Indeed, using thismethod, we are able to predict the stress conditions in hypothetical build scenarios.As shown in Fig. 3.9, this section focuses on the investigation of two types of FGM thin-wallstructures: vertical- and horizontal-dominant FGM walls. Vertical-dominant walls are characterizedby full contact between material A and the substrateâs top surface, with inclined material transitionregions defined by an angle θ1 (ranging from â90â¦ to 90â¦) between the material gradient directionand the build direction. On the other hand, the horizontal-dominant walls are fabricated when thesubstrateâs top surface contacts with the two base materials and the material transition region. Anangle θ2 (with the same definition as θ1, ranging from 0â¦ to 180â¦) is used to describe the tiltedmaterial transition region of these walls.Figure 3.9: Two DED deposition strategies for fabricating FGM thin-wall structures featuredby titled material composition transition regions: (a) Vertical- and (b) Horizontal-dominant FGMwalls.663.4. Bulk Residual Stress in DED-built FGM Structures Featuring Adapted Transitional RegionsFigure 3.10: Effects of material transitional angles on the bulk residual stress distributions ofvertical-dominant walls under θ1 = â2.4â¦, θ1 = 0â¦, and θ1 = 2.4â¦, and horizontal-dominant wallsunder θ2 = 45â¦, θ2 = 90â¦, and θ2 = 135â¦.Fig. 3.10 shows the distributions of longitudinal residual stress (Ïrexx) and their profiles atthe mid-length locations in vertical-dominant FGM walls at varying θ1 angles, and in horizontal-673.5. Summarydominant FGM walls at different θ2 angles, respectively. These FGM walls share the same materialgraded levels (a 25 wt% reduction in one materialâs composition and a 25 wt% increase in the otherat each transitional region) and deposition parameters as the FGM sample in Section 3.3. Theresults suggest that with the increase of θ1 from -2.4° to 2.4°, the maximum tensile residual stressincreases in the vertical-dominant walls, which peaks at the location where the pure IN718 startsto deposit. This increase in the peak tensile residual stress is proportional to the accumulatedplastic strain at the peak residual stress location, which is generated due to the plastic deformationin the deposition of the following pure IN718 layers. In addition, a larger θ2 value results inmore compressive residual stress at the top surface of the horizontal-dominant FGM walls. As θ2increases from 45° to 135°, the weight ratios of SS316 increase by 50 wt% at the FGMâs top surface.According to [92], the thermal expansion coefficient of SS316 is larger than that of IN718. Therefore,due to the restricted thermal expansion of the top surface material caused by the mismatch inthermal expansion coefficients between SS316 and IN718, compared to the walls with a lowerθ2, more compressive residual stresses are accumulated at the FGMâs top surface in horizontal-dominant FGM walls with a higher θ2. Moreover, due to smoother material transitions betweenSS316 to IN718 in the build direction, horizontal-dominant FGM walls present fewer residual stressfluctuations than vertical-dominant walls at the selected paths along the build (z) direction.3.5 SummaryThis chapter enhances the analytical and practical model described in Chapter 2 to provide funda-mental insights into the bulk residual stress generation in functionally graded DED processes. Thevalidated model enables novel process planning strategies, allowing for âstress condition by designâto be contemplated. The key findings are summarized as follows:(1) The enhanced model captures the thermo-mechanical response of FGMs in the fabricationof SS316/IN718 thin-wall structures by powder-fed laser DED processes. In the prediction, fivematerial composition transition paths were evaluated to assess the material mixing properties onthe final residual stress distribution, with one path following the designed weight percentages ofSS316 and IN718 in each deposition layer and the other four paths determined by measuring theaverage weight ratios of four elements Fe, Ni, Nb, and Ti, respectively. Both experimental andpredicted residual stress profiles present compressive residual stresses close to the substrate, whichshift to tensile states and continue to increase, featuring fluctuations, toward the top surface of thestructure. These fluctuations are driven by the combined effects of plastic strain accumulation andstrain/stress relaxation (caused by remelting) in the deposition of subsequent layers.(2) NRMSE was used to quantify the deviation between measured and simulated residual stressprofiles. NRMSE value for the designed material transitioning path from SS316 to IN718 is 22.59%.When using actual weight percentages of SS316 and IN718 detected by EDS through quantifyingthe weight ratios of elements Fe, Ni, and Ti, the NRMSE values are lower at 21.55%, 21.56%, and21.21%, respectively. However, due to the smaller proportion of Nb in the SS316/IN718 structure683.5. Summarycompared to Fe or Ni, Nb-based predictions were more influenced by measurement noise, with thehighest NRMSE value of around 26.42%. Thus, for more accurate residual stress predictions inSS316/IN718 FGM structures fabricated by DED processes, it is recommended to use actual weightpercentages of the two materials in each layer, which are quantified by elements Fe, Ni, or Ti whenEDS is available. However, predictions based on designed ratios remain acceptable for industrialapplications where extremely high accuracy is less critical.(3) Based on the validated model, fabricating two types of titled FGM thin-wall structures(vertical- and horizontal-dominant walls) was studied, and the effect of material gradient angles onthe final residual stress distribution was investigated. The results demonstrate that due to the mis-match of physical properties between the two base materials, materials graded modes significantlyinfluence the residual stress pattern and magnitude within the FGM parts. In vertical-dominantFGM walls, the maximum tensile residual stress is proportional to the material gradient angles.Conversely, increasing the material gradient angles in horizontal-dominant walls leads to more com-pressive residual stress at the top region of the FGM structures. This exploration provides insightsfor designing innovative DED strategies to satisfy the residual stress requirement in the fabricationof FGM components.69Chapter 4Surface Residual Stress in WetMachining Processes4.1 Introduction and Literature ReviewHybrid AM+machining applications mainly focus on high-strength materials (titanium- and nickel-based alloys), in which the use of cutting fluids is usually recommended. According to Shokrani etal. [144], the application of cutting fluids has the benefits of extending tool life, accelerating chipbreakages, and reducing the temperature in the cutting zones. Therefore, in order to understandthe final surface residual stress, this chapter investigates the wet machining in only-machiningoperations, focusing on the high-strength nickel-based alloys, that are widely used in the hybridAM and machining applications.Figure 4.1: A schematic diagram of wet cutting processes.As suggested in [4], in the machining of metallic components, cutting fluid (shown in Fig. 4.1)is usually supplied to the tool rake face (Nozzle 1) and/or the flank face (Nozzle 2) to achievelubrication and cooling. However, the use of cutting fluid has been criticized for its impact on theecology and its recycling and disposal costs. More importantly, Sharman et al. [143] and Devillez etal. [40] reported experimental results regarding the impact of cutting fluid application on surfaceresidual stress during the machining of high-strength materials. Therefore, investigation of the704.1. Introduction and Literature Revieweffect of the cutting fluid on the surface residual stress is essential to ensure that the fluid can beapplied effectively and efficiently.Liao et al. [99] concluded that surface residual stress plays a significant role in evaluating thecorrosion resistance and fatigue strength of machined components. The experimental results alsosuggest that the use of cutting fluid affects the residual stress distribution on the finished surface inthe wet machining of nickel-based alloys. Sharman et al. [143] experimentally studied the effects ofcutting fluid supply pressure and direction on the surface residual stress when turning Inconel 718.The results suggested that applying cutting fluid to the rake face resulted in higher tensile residualstress; however, the surface tensile residual stress decreased significantly when ultra-high cuttingfluid was applied at the flank face. Devillez et al. [40] measured and compared surface residualstresses after the dry and wet turning of Inconel 718. The results showed that the use of cuttingfluid reduced the surface tensile residual stress along the cutting velocity direction from around923 MPa to less than 300 MPa at a cutting speed of 40 m/min. However, when the cutting speedincreased to 80 m/min, the surface residual stresses were both tensile and with negligible differencein magnitude between dry and wet cutting. The above results show that the change of residualstress caused by cutting fluid depends on the cutting conditions, which determine whether themechanical or thermal load dominates the machined surface residual stress formation. When thethermal load is dominant, the final residual stress is mainly influenced by the cutting temperaturedistribution. In dry machining of titanium- and nickel-based alloys, Weng et al. [168] reportedthat the maximum cutting temperature on the tool rake face reaches around 800 °C, and Lianet al. [97] showed that the temperature variation between different cutting tool regions rangesup to 600 °C. The use of cutting fluid influences the transient temperature variations at both thecutting tool edges and the machined surfaces by carrying the heat away from the fluid-structureinterface. As a result, an obvious difference in machined surface residual stress between dry and wetcutting conditions can be observed. On the other hand, when the mechanical loads are dominantin generating the residual stresses, even though the cutting temperature drops significantly due tothe supply of cutting fluid, the difference in machined surface residual stress is negligible betweendry and wet cutting processes.Understanding the effect of cutting fluid in wet machining is important for tooling and processplanning in order to achieve the desired surface residual stress. As reviewed by Melkote et al. [69],numerical models have been developed to study the thermo-mechanical behavior of the workpiecematerials during machining and to reduce experimental trials for optimizing the cutting conditions.Among various numerical modelling techniques, FEM-based models are one of the most popularapproaches in metal cutting simulations. Such models can reveal the chip formation process, thetemperature, strain, and stress evolutions during machining, from which the material responseunder the addition of cutting fluid can also be studied.According to the convective heat transfer conditions, existing wet cutting simulations can beperformed using unconditional and conditional techniques. The unconditional technique assigns aconstant heat convection coefficient to the entire machined region regardless of the fluid-structure714.1. Introduction and Literature Reviewinteraction. Courbon et al. [33] developed a 2D FEM-based model with a constant heat convec-tion coefficient to simulate a high-pressure jet-assisted turning process of Inconel 718. The modelpredicted the cutting forces and chip formation under the cutting fluid pressure ranging from 30MPa to 130 MPa. The model was able to capture the reduction in tool-chip contact length as wellas the temperature drop on the cutting tool due to the cutting fluid application. Liu et al. [105]developed an FEM-based model with the same heat transfer coefficient to capture the residualstress profile in the cryogenic machining of Ti6Al4V. However, the unconditional assignment ofconvection conditions fails to account for the effect of cutting fluid on the transient evolutions oftemperature and stresses on the machined surface. Besides, the constant heat convection coefficientneglects the cutting fluid velocity, pressure, and the traveling distance from the fluid supply nozzlesto the machined surface.On the other hand, conditional heat convection relies on the fluid-structures (cutting tools andworkpieces) interaction properties. The corresponding models can be divided into sequentially andfully coupled thermo-mechanical analyses. Liu et al. [102] introduced a sequential, two-step modelwhere the coupled Eulerian-Lagrangian (CEL) method is applied to analyze the machining process,and the computational fluid dynamics is used to simulate the resulting cooling effect. However,the data transfer from the machining model to the cooling model is unidirectional, so the modelcannot capture the cooling effect induced by the cutting fluid during the machining process. Ayedet al. [13] proposed a 3D FEM-based model that simultaneously considers the chip formation andthe cutting fluid application. While the cutting fluid was modelled by the SPH/Eulerian method,the cutting tool and workpiece structures were described as the Lagrangian method. Moreover, auser-subroutine was defined to achieve heat exchange between the structure and the cutting fluid.Klocke et al. [81] introduced a 2D wet cutting model based entirely on the CEL method. In theirmodel, the chip formation process and the application of cutting fluid were simulated in the sameEulerian region, while only the cutting tool was modelled as the Lagrangian method. The modelwas later extended into 3D by Liu et al. [104]. Performed under various fluid supply pressures, the3D model captured the chip breakage due to cutting fluid application with experimental validation.The above literature review shows that the existing models focused on investigating the ap-plication of cutting fluid on the chip formation and breakage at the tool rake face side; while theeffect of cutting fluid on the machined surface residual stress has not been well understood. Toovercome the limitations in the existing models, this chapter develops a fully coupled CEL modelwhich simulates the effect of cutting fluid on the thermo-mechanical response of workpiece materi-als during wet cutting processes, by assigning one Eulerian region to include both workpiece andcutting fluid materials. Simulations of the actual cutting fluid flow from the nozzle to the cuttingregion are performed based on the Mie-Gru¨neisen Equation of State. As a result, the fluid velocityand flow property on the machined surface are used to consider its thermo-mechanical effect on themachined surface residual stress generation. For model validation, orthogonal cutting experimentsof Waspaloy are conducted under both dry and wet conditions. The measured cutting forces andresidual stress profiles via XRD techniques are compared with the predicted results.724.2. Wet Machining Model Development4.2 Wet Machining Model DevelopmentABAQUS is used to simulate both dry and wet cutting processes of Waspaloy. To obtain the surfaceresidual stress, as shown in Fig. 4.2, two stages need to be conducted sequentially:â¢ The tool/workpiece/fluid interaction, which consists of the simulation that includes the work-piece material deformation around the tool edge and the flow states of cutting fluid near thenewly generated surface;â¢ The cooling of the machined workpiece to room temperature to reach the thermal equilibriumcondition.Figure 4.2: A schematic diagram of wet cutting simulation.The machining simulation is performed based on the CEL method using ABAQUS/Explicitmode, and the cooling down simulation is based on the Lagrangian method using ABAQUS/Implicitmode. In order to achieve the data transfer between the two stages of the simulations, all the valuesof the physical parameters and the boundary condition information from the machining simulationsare first exported to a text file. The file is processed by modifying the label format to be adaptedto the cooling down model, and then is imported to the cooling down simulation as the initialcondition.4.2.1 Orthogonal cutting modelFig. 4.3 shows the initial geometry of the proposed wet machining model with the applied boundaryconditions. The FEM-based model is based on the CEL formulation. Specifically, the workpiece andcutting fluid materials are modelled using the Eulerian method, while the cutting tool is modelledby the Lagrangian method. According to the recommendation of Liu et al. [107], square elements734.2. Wet Machining Model Developmentwith an edge length of 5 µm are assigned to the tool-workpiece-fluid contact areas. These elementshave demonstrated sufficient accuracy in the prediction of chip formation and surface residual stress.In addition, the Eulerian region is further refined near the machined surface, i.e., the minimumelement size is defined as 2 µm in the uncut chip thickness (y) direction while the element sizeremains as 5 µm in the cutting speed (x) direction. The aim is to accurately identify the boundarylayer between the workpiece material and the cutting fluid after finishing the machining process.The rest of the Eulerian region is sparse in terms of the number of elements per unit area. Theelements near the tool edge are also assigned as 2 µm to maintain its circular shape. Analogue to[108], only one element is employed in the modelâs transverse direction, and the element width is setas 30 µm. Under the above meshing strategy, the Eulerian region shares 64,072 8-node thermallycoupled linear Eulerian brick elements (EC3D8RT), while the cutting tool has a total of 524 8-nodethermally coupled Lagrangian brick elements. The two x-y boundary planes of the FE model inthe z-direction are applied with a zero-displacement boundary condition to satisfy the plane straincondition, and the bottom surface of the Eulerian part is completely fixed during the machiningsimulation.Figure 4.3: Initial geometry and boundary conditions of the general wet cutting model (H = 0.12mm).At the start of the simulation, a proportion of the Eulerian region is filled with materials (theblue region marked as Aw is the workpiece material, and the region named as Af is the cutting744.2. Wet Machining Model Developmentfluid) while the rest is void. Then, the cutting tool advances into the fixed workpiece at a speedof Vc, and the cutting fluid flows towards the machined surface at a speed of Vf . Because of themovement of the cutting tool, the cutting fluid first fills the tool rake face region, and then achievescooling at the regions close to the tool rake and flank faces. During the wet cutting simulation, avariable Eulerian Volume Fraction (EVF) is employed to track the ratio of each material occupiedby an element on its volume as the element can be void (EVF = 0), partly filled, or fully filled byone or two materials (EVF = 1) during machining.Heat loss by forced convection occurs at the fluid-structure contact interface. However, heatloss through free convection (the heat transfer between the system and the environment) is nottaken into account because the machined surface will be fully immersed in the cutting fluid duringthe cutting process. Moreover, heat radiation is neglected throughout the simulation as the loss ofheat is negligible compared to heat conduction and forced convection in a very limited time period.Lastly, the cutting fluid evaporation is not taken into consideration.The following sections list the material models and the interaction properties used in the wetcutting simulation, as well as the procedures to determine Vf .4.2.1.1 Material modelsThe cutting tool is defined as a rigid body without considering tool wear and deformation. Themechanical and thermal properties of cutting fluid are simplified as water, due to the low concen-tration in the actually used cutting fluid. JC plasticity and damage models are used to capture theplasticity and damage behaviors of the workpiece materials, while the Mie-Gru¨neisen equation ofstate is selected to model the flow conditions of cutting fluid during the wet cutting process. Thematerial properties of cutting tool, workpiece, as well as cutting fluid are listed in Table 4.1, andthey are taken from Ayed et al. [13], Liu et al. [107] and Yang et al. [181].Table 4.1: Material properties of cutting tool (uncoated cemented carbide), workpiece (Waspaloy),and cutting fluid (water).Properties Cutting tool Workpiece Cutting fluidDensity [Kg·mâ3] 14,500 8,190 983Youngâs modulus [GPa] 696 213 (20 °C) -155 (900 °C) -Poissonâs ratio 0.25 0.33 -Thermal conductivity [WÂ·mâ1·°Câ1] 28.4 11.4 (20 °C) 0.598 (20 °C)25 (900 °C) 0.677 (100 °C)Thermal expansion [°Câ1] 5.2 1.18Ã 10â5 -(20 °C)1.6Ã 10â5 -754.2. Wet Machining Model DevelopmentProperties Cutting tool Workpiece Cutting fluid(900 °C)Specific heat capacity [JÂ·Kgâ1·°Câ1] 260 435 (20 °C) 4,182710 (900 °C) -Inelastic heat fraction - 0.9 -Dynamic viscosity [NÂ·s·mâ2] - - 1Ã10â3 (20 °C)0.28Ã10â3 (100°C)Kinematic viscosity [m2·sâ1] - - 1Ã 10â4Thermal conductance [kWÂ·mâ2·°Câ1] - 20 (30 MPa) 85.4220 (300 MPa)500 (600 MPa)i. JC ModelsThe JC plasticity model [74], as expressed in Eq. (2.25) is selected to calculate the flow stressof Waspaloy during machining, which describes the flow stress ÏÂ¯JC as a function of the equivalentplastic strain ε¯p, the equivalent plastic strain rate ËÂ¯Îµp, as well as the temperature T . Moreover, theJC damage initiation model is employed to identify the onset of material damage during cutting.According to the definition in [75], failure occurs when the equivalent plastic strain exceeds a criticalvalue ε¯p0, JC, which is expressed as:ε¯p0, JC =[d1 + d2 · exp(d3 · ÏmÏÂ¯JC)]·[1 + d4 · ln(ËÂ¯ÎµpËÂ¯Îµ0)]·[1 + d5 Â·(T â T0Tm â T0)], (4.1)where Ïm is the mean principal stress, Tm is the melting temperature, T0 is the reference (room)temperature, and di (i = 1 â¼ 5) refer to the five coefficients of the JC damage initiation model.The values of the coefficients for the JC plasticity model and damage initiation model are adoptedfrom Yang et al. [181], as listed in Table 4.2 and 4.3, respectively.Table 4.2: Coefficients of the JC plasticity model for Waspaloy.AJC [MPa] BJC [MPa] nJC CJC mJC ËÂ¯Îµ0 [sâ1] Tm [°C] T 0 [°C]1200 1284 0.54 0.006 1.2 0.001 1356 20764.2. Wet Machining Model DevelopmentTable 4.3: Coefficients of the JC damage initiation model for Waspaloy.d1 d2 d3 d4 d50.11 0.75 -1.45 0.04 0.89ii. Mie-Gru¨neisen equation of stateABAQUS/Explicit includes the equation of state models to describe the hydrodynamics behav-ior of materials [63]. We employ a linear form of the Mie-Gru¨neisen equation of state to model anincompressible viscous and inviscid laminar flow governed by the Navier-Stokes equation of motionfor the cutting fluid. Taking the principal Hugoniot as a reference curve, as presented by Bennett[21], the Mie-Gru¨neisen equation of state defines the pressure p as a function of the density Ï andthe internal energy Em by:pâ pH = Î · Ï · (Em â EH), (4.2)where pH and EH are the Hugoniot pressure and Hugoniot energy, which are only dependent onthe materialâs density; and Î is the Gru¨neisen ratio, which can be written as:Î = Î0 · Ï0Ï, (4.3)where Î0 is the Gru¨neisen coefficient and Ï0 is the reference density.The Hugoniot energy EH is related to the Hugoniot pressure pH , which can be expressed as:EH =pH · ηnor2 · Ï0 , (4.4)where ηnor is the nominal compressive volumetric strain. ηnor defines the ratio between the shockvelocity Us and the particle velocity Up, expressed as:ηnor =UpUs= 1â Ï0Ï, (4.5)Asay et al. [12] fitted the Hugoniot data and expressed the Hugoniot pressure pH as:pH =Ï0 · c20 · Î·nor(1â s · ηnor)2 ,(4.6)where c0 is the sound velocity, and s is the slope of Us versus Up. c0 and s are used to define thelinear relationship between Us and Up, which follows:Us = c0 + s · Up, (4.7)With the above condition, the elimination of Î , pH , and EH in Eq. (4.2) yields774.2. Wet Machining Model Developmentp =Ï0 · c20 · Î·nor(1â s · ηnor)2 Â·(1â Î0 · ηnor2)+ Î0 · Ï0 · Em, (4.8)where Ï0c20 equals the elastic bulk modulus at a small nominal strain.Based on the above analysis, the cutting fluid is modelled to provide zero shear strength witha volumetric response dominated by p = Kεvol, where K = Ï0c20 and εvol (= ηnor) defines themeasured nominal volumetric strain. In accordance with the recommendation by Ayed et al. [13],the values for Î0 and s are assigned as zero.4.2.1.2 Interaction propertiesThe proposed FEM-based wet machining model involves the definition of the Eulerian-Lagrangian(the tool/workpiece and tool/fluid interactions) and the Eulerian-Eulerian (the fluid/workpiece in-teraction) contact formulations. ABAQUS/Explicit describes the contact properties between theEulerian instances (the workpiece or cutting fluid) and the Lagrangian structure (the cutting tool)through an enhanced immersed boundary method [63]. This contact algorithm enables the compu-tation and automatic tracking of the interface between the Eulerian materials and the Lagrangianstructures. In addition, the Eulerian-Eulerian contact interaction is defined as sticky with no slipsbased on the default ABAQUS/Explicit configuration. More details related to the definition of theinteractions are listed as follows.i. Tool-workpiece interaction: A friction model proposed by Zorev [192] is employed todescribe the sticking and sliding behaviors at the tool-workpiece interface. By limiting the maximumshear stress at the interface, this model separates the tool/chip contact length into sticking andsliding regions, and the friction coefficient is assigned as 0.5 [26] in both dry and wet cuttingconditions. This is because, in the wet cutting simulation, the tool-workpiece interface is hardlyreached by the cutting fluid at the beginning, and the tool rake and flank faces are in contactwith the cutting fluid where the tool-workpiece contact is not present. Therefore, the effect of thecutting fluid on the friction between the cutting tool and the workpiece is neglected in the model.The fraction of heat generation due to the friction work is set as 1.0, and the generated heatis separated equally between the cutting tool and the workpiece material. During cutting, heatconduction occurs between the cutting tool and the workpiece material. The thermal conductanceis assigned as a function of contact pressure since the conductance is not supported as a function ofclearance between the Lagrangian structure and the Eulerian material in ABAQUS/Explicit. Theconductance coefficient is adopted from Agmell et al. [1].ii. Tool-fluid interaction: The interaction between the cutting tool and the cutting fluidis assumed to be rough (the friction coefficient is specified as infinite), and no heat generationoccurs at the interface. The heat transfer between the tool and the fluid is modelled by thepressure-dependent thermal conductance coefficient. Specifically, once the nodal contact pressureat the cutting tool is greater than zero, heat conduction is performed from the tool/fluid with hightemperature to the fluid/tool with low temperature. This conductance coefficient is adopted from784.2. Wet Machining Model DevelopmentLiu et al. [102], which represents the average heat exchange coefficient on the tool rake face in thewet cutting process of nickel-based alloys.iii. Fluid-workpiece interaction: In ABAQUS/Explicit, Eulerian materials contact eachother in a sticky behavior without any slips at the interface [63]. Thermally coupled Eulerianelements allow multiple materials with different thermal properties and support heat transfer. Tosolve the heat transfer equation for multiple materials in an Eulerian element, the thermal properties(such as thermal conductivity and specific heat capacity) are defined as volume-averaged based onthe rule of mixtures. For example, λavg represents the average thermal conductivity in an Eulerianelement occupied with two materials, which can be calculated as:λavg = EVFw · λw + EVFf · λf , (4.9)where λw and λf define the thermal conductivity of the workpiece material and the cutting fluid,respectively; EVFw and EVFf are the fractions of workpiece material and cutting fluid occupiedwithin one Eulerian element (0 â¤ (EVFw + EVFf ) â¤ 1), respectively.4.2.1.3 Determination of fluid velocity VfAccording to the measured flow rate and the dimension of the nozzle, the velocity of cutting fluidat the nozzle outlet can be determined (which is calculated as 6,400 mm/s for the machine usedin model validation). After flowing out from the nozzle, the cutting fluid needs to travel a certaindistance before reaching the machining zones. Since the wet cutting simulation takes place in afew milliseconds, it is not necessary to model the entire flow state from the nozzle outlet to theworkpiece. However, the loss of fluid velocity must be included in the wet cutting model.Liu et al. [103] concluded two sources of fluid velocity loss during the wet cutting process, knownas the loss caused by air resistance and the loss due to the change of flow state. This work onlyconsiders the second source because of the transformation in the flow regime, while the air-fluidinteraction is neglected due to the limited distance between the nozzle and the machined surface.Fig. 4.4 (a) depicts the flow state of cutting fluid after jetting out from a nozzle with a diameterof D at a velocity of Vf . The cutting fluid immediately changes into the turbulent round jet flowafter pumping out from the nozzle. As the cutting fluid contacts the machined surface, it graduallychanges to lamellar flow. However, the presence of a cutting tool interrupts the above flow andcreates different paths for the fluid to follow. A portion of the cutting fluid squeezes into the limitedspace between the tool flank face and the newly generated machined surface. To fit into this area,this portion of cutting fluid transforms into a turbulent flow, featured by vortices behind the cuttingtool, as shown in Fig. 4.4(b). Another portion of the cutting fluid is situated above the turbulentvortex, remaining as the phase of laminar flow. Finally, this portion of cutting fluid spreads outfreely after moving away from the constraints of the cutting tool. As the cutting tool moves further,the flow state above the machined surface is gradually dominated by the lamellar flow. The lamellarflow near the machined surfaces helps in cooling the machined surface. The cutting fluid in this794.2. Wet Machining Model Developmentstate features a velocity gradient above the fluid-workpiece interface. Due to the viscosity of thecutting fluid, the cutting fluid velocity near the machined surface remains zero. Let Vfw denotethe maximum velocity of the cutting fluid in the laminar flow zone, which determines the amountof heat removed by the cutting fluid from the machined surface. In addition, the difference in themagnitude between Vf and Vfw represents the fluid velocity loss due to the change of flow statefrom the cutting fluid nozzle to the machined surface.Figure 4.4: Flow states of cutting fluid in wet cutting: (a) Schematic diagram of flowing states;(b) Simulated flowing conditions (Vc = 86 m/min and H = 0.06 mm).The proposed wet cutting model is modified to determine Vfw and then to quantify the lossof the fluid velocity at a given cooling condition. Fig. 4.5(a) shows the initial geometry andboundary conditions of the modified wet machining model. In this model, the movement of thecutting tool is neglected. The cutting fluid jetting area has a length of 4 mm which is equal to theactual diameter of the fluid supply nozzle. The thickness of the workpiece material is reduced to 1mm. Other boundary conditions remain the same as the wet cutting model. The Eulerian regionnear the machined surface is locally refined (elements with a length of 0.5 mm). Coarser gridswith a maximum element length of 1 mm are assigned in the remaining Eulerian areas to improvecomputational efficiency. Under the traveling distance Ï of 120 mm, the Eulerian part contains19,500 EC3D8RT elements.804.2. Wet Machining Model DevelopmentFigure 4.5: Fluid velocity prediction model and simulation results: (a) Initial geometry andboundary conditions; (b) Resultant velocity distribution (Vf = 6400 mm/s and Ï = 120 mm).Fig. 4.5(b) shows the velocity distribution of cutting fluid after reaching the stable state, wherethe traveling distance Ï is 120 mm and the cutting fluid velocity Vf is 6,400 mm/s. The fluidvelocity is extracted from Path 1, and the result is presented in Fig. 4.6(a). It can be seen thatthe fluid velocity increases rapidly from zero at the fluid-workpiece interface to a stable value nearthe top surface of the fluid. The stable value equals Vfw, which determines the heat convectioncoefficient on the machined surface. A point shown in Fig. 4.5(b) is selected to record Vfw. Asshown in Fig. 4.6(b), Vfw remains unchanged at around 5,680 mm/s after the traveling distanceexceeds 60 mm. The results indicate that the loss of cutting fluid velocity due to the change of flowstate is constant after reaching a critical distance. In the wet cutting experiment, the measuredcutting fluid traveling distance is around 135 mm. Based on the above analysis, the maximumcutting fluid velocity on the machined surface is 5,680 mm/s, which will be set as Vf in the wetcutting model for the following surface residual stress prediction.814.2. Wet Machining Model DevelopmentFigure 4.6: Simulation results of cutting fluid velocity (a) along Path 1; (b) at the selected point.4.2.2 Cooling down modelTo obtain the residual stress on machined surfaces, as suggested by Liu et al. [107], the machinedsurface needs to be cooled to room temperature after completing the machining process. Moussaet al. [121] reported that the computational time varies from weeks to months when cooling theEulerian structure in ABAQUS/Explicit. In this study, the cooling down process is conductedbased on a Lagrangian method in an implicit mode, which can save the calculation time from daysto minutes. Fig. 4.7 shows the dimensions and boundary conditions of the cooling down model.This model has 15,600 8-node thermally coupled Lagrangian brick elements (C3D8RT), which sharethe same material properties as the wet cutting model. The meshing strategy inherits from thecutting model, and all the values of physical parameters and boundary condition information areimported from the cutting model.Figure 4.7: Cooling down model with the application of boundary conditions.During cooling down, heat radiation is neglected while heat convection only occurs between themachined surface to the environment. A constant heat convection coefficient was assumed in the824.3. Experimental Setup and Proceduresmodel to simplify the complex heat transfer behaviors at the air/fluid-workpiece interfaces. In drycutting, the convection heat transfer is assumed to be a free convection condition. According toSartori [137], the free convection coefficient is set as 10 W/(m2·â). The application of cutting fluidschanges the free heat convection into a forced convection. The forced heat convection coefficient isdependent on the cutting fluid velocity and properties. Given the cutting fluid properties, Holman[65] described the average forced heat convection coefficient hconv on the machined surface with alength of Lsurf , which is calculated as:hconv = 0.664 · kfLsurf· Re1/2Lsurf · Pr1/3, (4.10)ReLsurf =Vf · Lsurfνcf, (4.11)Pr =Cpf · µfkf, (4.12)where ReLsurf is the Reynolds number; Pr is the Prandtl number; kf , Cpf , νcf , and µf arethe thermal conductivity, specific heat capacity, kinematic viscosity and dynamic viscosity of thecutting fluid, respectively. Please note that the cutting fluid properties used in the calculation aretemperature-independent. The values are adopted from Ayed et al. [13], which are defined as thecoefficients corresponding to room temperature (20 â). This is due to the continuous supply ofcutting fluid from the nozzle and the occurrence of turbulence near the machined surface. As aresult, the heated cutting fluid is immediately replaced by new and unheated fluid after contactingthe machined surface. Under the cutting fluid velocity Vf of 5,680 mm/s obtained from the cuttingfluid simulation in Section 4.2.1, hconv is calculated as 4,050 W/(m2·â).4.3 Experimental Setup and ProceduresThe workpiece material is AMS 5704 Waspaloy (HRC 41.8) after solution annealing at 996-1,038°C for 4 h, oil quenching, stabilizing at 843 °C for 4 h, air cooling, aging at 760 °C for 16 h, andfinally air cooling.Radial turning experiments of Waspaloy were conducted in a CNC turn-mill machine center(Mori Seiki NT 3150 DCG 1000 S) under both dry and wet cutting conditions. During wet cuttingprocesses, the machining center supplied 6 bars of cutting fluid through nozzles with a diameter of4 mm. The actual flow rate of cutting fluid coming out from the nozzle was measured as around4.8 L/min using a plastic measuring cup. Water-based cutting fluid with a concentration of 6.5%was used in the experiments.The workpieces were hollow cylinders with a maximum diameter of 180 mm and a thickness of 20mm. The cutting tools were uncoated carbide inserts, with the geometric parameters listed in Table4.4. A laser confocal microscope was used to measure the tool edge radius (re), with the final valueobtained by averaging the measurements of 5 tool edge profiles. During experiments, the inserts834.3. Experimental Setup and Procedureswere mounted on a left-handed tool holder (made of stainless steel 316), and they were assembled ona force dynamometer (Kistler 9121). As shown in Fig. 4.8, an L-shape adapter (made of aluminumalloy 6061) was designed to fix the dynamometer to the milling spindle. The cutting parametersused in the simulations and the experiments are summarized in Table 4.5. Specifically, the cuttingspeed (Vc) was fixed at 86 m/min while the cooling condition and the feed rate (uncut chip thickness,H) varied in the experiments. Two radial turning tests were conducted at each cutting condition.The first cut had a depth of cut (DOC) of 0.75 mm. In this case, the whole machined surface wasgenerated by the tool nose. The second cut was performed under a larger DOC of 1.5 mm. Let wdenote the width of machined surfaces finished by straight cutting edges. In the second cut, w is0.75 mm, which is equal to the width of cut in the orthogonal cutting condition. Under the abovecutting conditions, the machined surface finished by the straight cutting edge meets the dimensionrequirements for the following residual stress measurement. Three cutting force components weremeasured in each cut, including tangential (Ft), feed (Ff ), and radial force (Fr), and the resultswere used for the model validation. In addition, each experiment was performed with a new insertto avoid tool wear.Table 4.4: Cutting tool geometric parameters used in the experiments.Parameters ValuesRake angle [°] 0Clearance angle [°] 7Tool edge radius re [µm] 20± 0.5Tool nose radius rn [µm] 750Table 4.5: Experimental conditions.No. Vc [m/min] H [mm] DOC [mm] Cooling conditions1 86 0.06 0.75 Dry2 86 0.06 1.50 Dry3 86 0.06 0.75 Wet4 86 0.06 1.50 Wet5 86 0.12 0.75 Dry6 86 0.12 1.50 Dry7 86 0.12 0.75 Wet8 86 0.12 1.50 Wet844.3. Experimental Setup and ProceduresFigure 4.8: Experimental setup, schematic diagrams of wet cutting process, and tool radiusmeasurements.After finishing the machining process, as shown in Fig. 4.9, surfaces generated by the largerDOC were separated into several pieces by WEDM for residual stress measurements and surface854.3. Experimental Setup and Proceduresmicrostructure observations. The dimensions of the WEDM cut samples are listed in Fig. 4.9.A minimum distance of 2 mm was kept between the sample edge and the X-ray beam boundaryto avoid the change of machining-induced stress status due to residual stress superposition andrelaxation caused by the WEDM process. The cut samples were placed in an X-ray diffractometer(Proto LXRD) to measure the residual stress in the cutting velocity direction, and a polisher (Proto8818-V3) was used to electropolish samples to measure in-depth residual stress. In the measurement,the X-ray beam was adjusted to focus on the machined surface generated by the straight cuttingedges to ensure the orthogonal cutting configuration. The polishing was conducted at a distanceinterval of 5 â 20 µm, until 0.1 mm depth below the original machined surface, where the residualstress values converge to zero. More detailed parameters in the residual stress measurements arelisted in Table 4.6. Samples used for microstructure testing were first mounted in plastic moldsusing a 25:3 mixture of EpoFix Resin (Struers) and EpoFix Hardener (Struers). Before pouring thesolution, the molds were treated with Stoner E206 silicone mold release so that the parts can beeasily removed from the molds after solidification. Then, the samples were ground and polished bya manual grinder-polisher (EcoMet 30, Buehler). Silicon carbide waterproof metallurgical papers(Allied High Tech) were used to grind the exposed uneven surface of the sample. To remove thescratches induced by grinding, the ground surfaces were polished with diamond films (Struers).Water and water-based monocrystalline diamond suspensions served as lubricants in grinding andpolishing processes, respectively. The polished surfaces were then etched by Kallingâs No.2 reagentfor 20 seconds. After finishing the metallurgical treatments, the samples were placed in a confocalmicroscope (LEXT OLS3100) to capture the microstructure.Figure 4.9: Sample preparations for surface residual stress measurements and microstructureobservations.864.4. Results and DiscussionTable 4.6: Surface residual stress measurement conditions.Items DescriptionX-ray source Mn K alpha (5.9 KeV)Filter type No K beta filter used, confirmed no differencewith/without filterDiffraction plane {311}Bragg angle 152°Collimator size 0.5 mm Ã 2 mmExposure time 20 seconds (2 seconds Ã 10 exposures) at eachtilt angleΨ -tilt oscillation 11 tilts with Ψ up to 39°X-ray tube power 32 kV and 30 mAX-ray elastic constants (1/2)S2: 6.61Ã 10â6 MPaâ1,âS1: 1.42Ã 10â6 MPaâ14.4 Results and DiscussionIn this section, the simulated cutting forces and surface residual stress profiles are compared with theexperimental measurements. The comparison results are discussed, followed by detailed analyses ofthe coolant effects on the contact stress, temperature, as well as plastic stress and strain evolutions.4.4.1 Model validation4.4.1.1 Cutting forcesCutting forces reflect the thermo-mechanical loads applied on the machined surfaces by the cuttingtool. For comparison purposes, force per width (N/mm) is adopted. Specifically, in the simulation,a reference point on the tool is selected to output the predicted cutting forces. The obtained resultsare then divided by the width of elements for comparison. Fig. 4.10 shows one set of time-varyingcutting forces under the given cutting conditions. The periodic variation of both tangential (Ft)and feed forces (Ff ) is attributed to the formation of chip segmentation.874.4. Results and DiscussionFigure 4.10: Cutting forces predicted by the numerical model during wet cutting of Waspaloyunder conditions of Vc = 86 m/min and H = 0.06 mm.In the experiments, there are two groups of force measurements in each cutting condition. Theresults used for validation can be obtained by subtracting the measured forces at 0.75 mm DOC(same as the tool nose radius) from the measured values at 1.5 mm DOC to exclude the tool nosewith curved geometry, and then dividing the involved length of straight cutting edges (0.75 mm).The purpose of performing the subtraction at different widths of cut is to ensure the orthogonalcutting configuration. This is because, in the longitudinal turning experiments, a portion of thecutting forces are generated due to the contact between the tool nose and the workpiece, which isan oblique cutting configuration due to the round tool nose geometry. To remove this effect, thecutting forces obtained by the smaller depth of cut (only the tool nose is involved in the cutting) aresubtracted from the measured forces under the larger depth of cut (both the tool nose and straightcutting edge are involved in the cutting). As a result, the calculated results are only contributedfrom the straight cutting edge, which is an orthogonal cutting configuration. This experimentalprocedure to obtain the cutting forces in an orthogonal cutting configuration was reported by Jinand Altintas [72]. The uncertainties of the above procedure originate from the inconsistency of toolnose wear under different widths of cuts. This is based on the fact that the cutting condition witha larger width of cut may generate more severe tool wear due to larger chip load and cutting forces.By subtracting the cutting forces obtained by inserts with slight tool nose wear from the resultsobtained by the inserts with severe tool nose wear, the cutting forces generated by the straightcutting edge might be underestimated. However, in our experiments, fresh inserts were used in allexperiments to minimize this effect, as only initial tool wear was involved.Fig. 4.11 lists typical experimental force profiles in the time domain when machining Waspaloy.To exclude the effect of machine tool vibration, the measurements are filtered by a fourth-orderButterworth low-pass filter with a cutoff frequency of 50 Hz. Measured radial cutting forces arenot included under orthogonal cutting configuration.884.4. Results and DiscussionFigure 4.11: Measured and filtered cutting forces (Vc = 86 m/min, H = 0.06 mm, wet cutting):(a) DOC = 0.75 mm; (b) DOC = 1.5 mm.Fig. 4.12 compares the average cutting force components between experiments and simulationsunder both dry and wet cutting conditions. It can be found that tangential forces obtained fromthe simulations show good agreement with the experimental measurements (mean squared error:1.34%), whereas the feed forces are underestimated (mean squared error: 7.18%). Liu et al. [107]used CEL techniques to simulate orthogonal cutting processes of nickel-based alloys, and reportedthat the simulated feed forces were underestimated by over 40% compared to the measured data.Yameogo et al. [178] included recrystallization in simulating the machining of Ti6Al4V, and showedthat the simulated feed forces were less than 85% of the experimental data. In our proposed FEM-based model, the maximum deviation in feed force occurs at an uncut chip thickness of H = 0.12mm, showing an underestimation of around 38% compared to measurements. The underestimationof the feed force is attributed to a more simplified friction model compared to the complex tool-chipinteraction phenomenon. Moreover, even if a new insert was used in the experiment, the machiningprocess involves rapid tool edge geometry change due to initial tool wear, which contributes tothe increase of the feed force. As reported by Laakso et al. [88], the initial flank wear contributes20%-50% of the feed force, even without considering the friction. This contribution fluctuates based894.4. Results and Discussionon various cutting parameters, such as the cutting speed and the uncut chip thickness. Segreto etal. [139] performed turning experiments of Inconel 718 and found that the initial flank tool wearrate increases with the cutting speed and the uncut chip thickness. Therefore, the larger differencebetween the simulations and the experimental results at 0.12 mm uncut chip thickness is becauseof the higher initial tool wear rate caused by a larger cutting force at a higher chip load. This workfocuses on the effect of cutting fluid on the surface residual stress; therefore, the tool wear effect isnot included in the simulations.Figure 4.12: Comparison of average tangential (Ft) and feed (Ff ) forces between experimentsand simulations under Vc = 86 m/min (a) H = 0.06 mm; (b) H = 0.12 mm.4.4.1.2 Surface residual stressTo further validate the wet cutting model, the predicted surface residual stress in the cuttingvelocity direction (S11) is compared with measured values. Fig. 4.13 shows surface residual stressdistribution after the orthogonal cutting of Waspaloy with Vc = 86 m/min and H = 0.06 mm. Itis noted that surface residual stress is periodically distributed within the machined workpiece dueto varied thermo-mechanical loads, which are associated with the chip segmentation, applied bythe cutting tool. To enhance the reliability of the predicted residual stress profile, as shown inFig. 4.13, in one period within the stable zone, at least 15 paths are chosen to output the surfaceresidual stress profiles. This is based on the principle that the periodic variation of the surfaceresidual stress spans the whole length in the cutting velocity direction. Therefore, the averagevalue from one variation period represents the mean value of the surface residual stresses for thearea at the size of the collimator. Meanwhile, instead of averaging the residual stress results basedon the penetration depth of the X-ray source to obtain the in-depth residual stress profile, thesimulated residual stresses are averaged at each nodal position. This is because the penetrationdepth of the X-ray source is unmeasurable, and it varies when moving the measurement locationas summarized by Fitzpatrick et al. [49]. This variation can arise from the inhomogeneity of thematerial properties and the non-uniform reflectivity of the measured surface after electropolishing.904.4. Results and DiscussionFigure 4.13: Surface residual stress distribution in the cutting velocity direction (Vc = 86 m/minand H = 0.06 mm) and selected paths: (a) Dry cutting; (b) Wet cutting.Fig. 4.14 compares the residual stress profiles obtained from measurements and simulationsafter the orthogonal cutting of Waspaloy. The measured residual stress results showed that themachined surfaces present tensile residual stress under the given cutting conditions. Meanwhile,under the cutting speed of 86 m/min, the use of cutting fluid reduces the machined surface tensileresidual stress from 1582 MPa to 1232 MPa at H = 0.06 mm and from 1898 MPa to 1770 MPaat H = 0.12 mm, respectively. In addition, results also suggest that predicted residual stressprofiles match with the measured values under the given cutting conditions. The differences in themeasured and predicted surface tensile stresses range from -13.6% to 1.9%. Furthermore, both theexperimental and predicted results reveal the same trend regarding the impact of cutting fluid onthe surface residual stress. For instance, when Vc = 86 m/min and H = 0.06 mm, the use of cuttingfluid in experiments leads to a reduction of surface tensile residual stress by around 350 MPa, whilethe mitigation in prediction is about 254 MPa.Different sources of errors affect the prediction results and the experimental measurements. Atfirst, the accuracy of the experimentally measured residual stress profiles can be influenced by twoerror sources. The first is the WEDM-induced residual stress relaxation in the sample preparationprocedure. To quantify the residual stress relaxation in the WEDM process, a 2D FEM-basednumerical model is developed, with the modelling details presented in Appendix D. The simulationresults suggest that under the given cutting conditions, the surface residual stress relaxation dueto the WEDM accounts for around 15.9 % of the original residual stress in the cutting velocitydirection. In addition, when measuring residual stress through the XRD technique, potential sourcesof measurement error also include the divergence of the X-ray beam and the occurrence of strain914.4. Results and Discussionrelaxation in the layer removal process. The above two sources of errors are included in Fig. 4.14by including error bars in the experimental results. Moreover, inaccuracies in predicted residualstress profiles arise from the material constitutive parameters, the change of tool geometry due togradual tool wear, and the neglect of the residual stress variation caused by phase transformation.Specifically, the neglection of the tool geometry change due to initial tool wear and the simplificationof the frictional coefficient causes the underestimation of the cutting forces, especially the feed force.As a result, in cases where mechanical loads dominate the formation of residual stress, the finalcompressive residual stress will be underestimated. However, under the given cutting conditionsin this study, all the machined surfaces present tensile residual stress, and the final tensile residualstress values on the machined surface depend on the thermal shrinkage of the workpiece materialduring cooling down. Therefore, even though the feed forces are underestimated, the proposedmodel still has the ability to capture the tensile residual stress on the machined surfaces withacceptable accuracy.Figure 4.14: Comparison between experimentally and numerically obtained residual stress profilesin cutting velocity direction with Vc = 86 m/min (a) H = 0.06 mm, dry cutting; (b) H = 0.06 mm,wet cutting; (c) H = 0.12 mm, dry cutting; and (d) H = 0.12 mm, wet cutting.924.4. Results and DiscussionOverall, the comparison results demonstrate acceptable agreement and suggest that the pro-posed modelling procedure is capable of simulating wet cutting process while generating reliablecutting forces and surface residual stress profiles. In the following sections, the simulation results(local normal/shear contact stress and workpiece temperature) are used to further explain the cut-ting fluid effect on the mechanical and thermal loads exerted on the machined surface. In light ofthese findings, the governing mechanism of surface residual stress formation in wet cutting processesis elucidated.4.4.2 Effect of cutting fluid on local normal/shear contact stressesThe machining-induced compressive residual stress is associated with the mechanical effect (toolcompression followed by tension) that the cutting edge below the stagnation point exerts on themachined surface. The quantification of the mechanical effect can be achieved by evaluating thelocal normal and shear contact stresses on the tool edge-workpiece contact area. This sectioncompares the local normal and shear contact stresses on the tool edge in dry and wet cutting ofWaspaloy. The aim is to quantify the difference in mechanical loads applied by the cutting tool onthe machined surface during dry and wet cutting processes.To evaluate the local normal and shear contact stresses, it is required to determine the stagnationpoint on the tool edge. Fig. 4.15 shows the flow of materials around the tool edge during theorthogonal cutting of Waspaloy. Point S refers to the stagnation point where the direction of thematerial flow velocity bifurcates. After the stagnation point is determined, the local normal andshear contact stresses are calculated by summing the normal (CNORMF) and shear (CSHEARF)contact force per unit width at all the tool edge nodes in the area SB, and then dividing them bythe corresponding contact area, respectively. Fig. 4.16 (a) and (b) represent the distribution ofcontact forces CNORMF and CSHEARF on the tool-workpiece interface during the wet cutting ofWaspaloy, respectively. It can be seen that CNORMF reaches its maximum value at the currentstagnation point, while CSHEARF becomes zero. However, as shown in Fig. 4.16 (c), due to theformation of serrated chips, the local normal and shear contact forces vary periodically. Therefore,to enhance the reliability, the average values of local normal and shear contact force are obtained forone complete chip segment period. The average local normal/shear contact stresses are calculatedand compared during the dry and wet cutting of Waspaloy, as presented in Fig. 4.17. It is shownthat although the measured cutting forces dropped by around 8.6 % with the use of cutting fluid,the average normal/shear contact stress in wet cutting remains the same as that in dry cutting. Theresults suggest that the impact of cutting fluid on the local mechanical load is negligible, therebygenerating the same compressive residual stress as dry cutting.934.4. Results and DiscussionFigure 4.15: Materials flow around the tool tip during orthogonal cutting of Waspaloy: (a) Drycutting; (b) Wet cutting.Figure 4.16: Contact forces at the tool-workpiece interface (Vc = 86 m/min, H = 0.06 mm, wetcutting): (a) CNORMF; (b) CSHEARF; (c) Contact force evolution at the point S.944.4. Results and DiscussionFigure 4.17: Average local contact stress when orthogonal cutting of Waspaloy under Vc = 86m/min and (a) H = 0.06 mm; (b) H = 0.12 mm.4.4.3 Effect of cutting fluid on machined surface temperature distributionThis section quantifies the effect of cutting fluid on machined surface temperature distributionduring orthogonal cutting and cooling down processes. Fig. 4.18 (a) and (b) show the temperaturedistribution on the machined surfaces during the orthogonal cutting of Waspaloy under dry and wetcutting conditions, respectively, with Vc = 86 m/min and H = 0.06 mm. The results suggest thatthe temperature on the machined surface reduces more rapidly with the application of cutting fluidcompared to the machined surface in dry cutting. This is attributed to the forced heat transfermechanism, which is driven by the flow of cutting fluid that effectively takes heat away from thefluid-structure contact interface. In addition, the temperature difference of the machined surfacenear the tool edge is negligible under dry and wet cutting conditions. This is because the cuttingfluid may not have direct contact with the tool edge and the machined surface at the tertiary zone,due to the flow pattern and the relative tool-workpiece motion in the simulated machining condition.Fig. 4.19 shows the temperature distribution of the workpiece material along the horizontal andvertical directions close to the tool edge, in order to understand the mechanism that drives thetemperature difference between dry and wet cutting. Under the uncut chip thickness of H =0.06 mm, Fig. 4.19 (a) compares the temperature distribution along a vertical Path 1 under bothcutting conditions. These distributions correspond to the heat generated by plastic deformationand tool-workpiece friction during machining. As the two cutting processes generate almost thesame temperature distributions, the use of cutting fluid does not affect the heat generation nearthe tool edge. In contrast, Fig. 4.19 (b) compares the temperature distributions on a horizontalline Path 2. The two temperature distributions exhibit sizeable differences (around 60 °C) after themachined surface is fully immersed by cutting fluid (the distance away from the tool edge is morethan 0.5 mm). Under dry cutting, the temperature distribution is only caused by heat conductionfrom the machined surface to the workpiece. Under wet cutting, however, the distribution isdetermined by both conduction and convection. This relatively smaller difference is primarily due954.4. Results and Discussionto the lower flow rate of the cutting fluid we used in the experiments, which is around 4.8 L/min.As a result, much of the heat generated on the machined surface during wet cutting is conductedinto the subsurface material before contact with cutting fluid, resulting in limited reduction of themachined surface temperature due to the application of cutting fluid. A higher flow rate (usuallyaround 10 L/min in industrial applications) could potentially enhance the cooling effect and yielda larger temperature difference. Although experimental validation of the temperature fields underboth dry and wet conditions was not conducted in this work, the predicted surface residual stressprofile aligns well with the measurements. In the future, the influence of coolant flow rate on thefinal machined surface temperature and residual stress will be studied in the machining of difficult-to-cut materials. Besides, Fig. 4.19 (c) and (d) show that the temperature distributions follow thesame trend when the uncut chip thickness changes to 0.12 mm under both dry and wet orthogonalcutting conditions.Figure 4.18: Temperature distribution during orthogonal cutting processes in the case of Vc = 86m/min, H = 0.06 mm. (a) Dry cutting; (b) Wet cutting.964.4. Results and DiscussionFigure 4.19: Workpiece temperature distribution during orthogonal cutting processes at Vc = 86m/min. (a) Path 1, H = 0.06 mm; (b) Path 2, H = 0.06 mm; (c) Path 1, H = 0.12 mm; (d) Path2, H = 0.12 mm.Fig. 4.20 (a) and (b) represent the machined workpiece temperature distributions after thesimulations of the cutting processes are complete and when the machined workpieces are cooleddown for 0.01 s at Vc = 86 m/min and H = 0.06 mm. To further quantify the effect of cuttingfluid on the machined surface temperature during the cooling down process, the two paths whichshare the same locations and origins as Fig. 4.18 are selected on the machined workpiece, withthe temperature results shown in Fig. 4.21. Fig. 4.21 (a) compares the temperature distributionalong Path 1. The results demonstrate an obvious temperature difference of 50 °C on the machinedsurface between dry and wet cooling processes. The above difference diminishes significantly as thedepth increases, becoming less than 10 °C at the depth of 0.05 mm beneath the machined surface.Moreover, the peak temperature occurs on the machined surface during the dry cooling condition,while it is 0.02 mm below the machined surface under the wet cooling condition. These differencesarise from the amount of heat transferred through heat convection. In dry cutting conditions, the974.4. Results and Discussiontemperature decrease of the machined surface in the cooling down process results from the heatconduction to the lower temperature and the heat convection to the surrounding environment.However, using cutting fluid increases the amount of heat transferred to the environment due toforced heat convection. At the same time, the cooler machined surface drives the heat generated onthe subsurface due to plastic deformation to be transferred to the surrounding materials throughheat conduction. As a result, the machined workpiece temperature along Path 1 is lower in wetcooling conditions compared to dry cutting, while the maximum workpiece temperature occurs atthe subsurface in the wet cooling condition. The same trend can also be captured at the uncut chipthickness of 0.12 mm under both dry and wet cooling conditions, as shown in Fig. 4.21 (c). Similarto the temperature distributions on the machined surface during dry and wet orthogonal cuttingprocesses, the machined surface temperature (Path 2) during wet cooling conditions remains lowerthan that of the dry cooling conditions with a temperature difference of around 60 °C, as shown inFig. 4.21 (b) and (d).Figure 4.20: Workpiece temperature distribution during cooling down processes in the case of Vc= 86 m/min, H = 0.06 mm. (a) Dry cutting; (b) Wet cutting.984.4. Results and DiscussionFigure 4.21: Workpiece temperature distribution during cooling down processes under Vc = 86m/min. (a) Path 1, H = 0.06 mm; (b) Path 2, H = 0.06 mm; (c) Path 1, H = 0.12 mm; (d) Path2, H = 0.12 mm.4.4.4 Effect of cutting fluid on machined surface residual stressIn dry cutting, the final profile of machined surface residual stress depends on the non-uniformplastic deformation of workpiece materials caused by mechanical and thermal loads. To understandthe formation mechanism of surface residual stress when machining Waspaloy with the cutting fluidapplication, Fig. 4.22 compares the nodal temperature history (NT11), stress evolution along thecutting velocity direction, and plastic strain formation along the cutting velocity direction (PE11)at a location on the machined surface during both dry and wet cutting processes. The stress andstrain evolutions at other locations follow the same mechanism as Fig. 4.22 in steady-state cutting.According to the simulated nodal temperature, plastic stress and strain evolutions in wet cuttingcan be divided into two stages: before and after the contact with the cutting fluid. In the firststage, with the advancement of the cutting tool, the plastic stress rapidly reaches the maximum994.4. Results and Discussioncompressive value and then gradually transforms into tensile stress. The compressive stress iscaused by the tool edge on the workpiece material ahead (mechanical effect) and the constrainedthermal expansion of workpiece material around the tool edge (thermal effect); the tensile stresscomes from the traction force by the flank face on the newly machined surface (mechanical effect)and the thermal contraction of surface material (thermal effect). Meanwhile, the plastic strainincreases significantly to tensile, and then gradually decreases after the tool moves away. Thisvariation is dominated by the thermal effect: the increase in tensile strain originates from therestricted thermal expansion of workpiece material around the tool edge; however, the decreasecomes from the thermal contraction of surface material after the cutting tool moves away. Theabove stress and strain curves are the same under dry and wet cutting conditions, since the cuttingfluid has no effect on the evolution of local contact stress below the stagnation point and heatgeneration at the tool-workpiece interface, as explained in Sections 4.4.2 and 4.4.3.Figure 4.22: Thermal history, plastic strain, and stress evolution at a surface point during or-thogonal cutting of Waspaloy: (a) Vc = 86 m/min, H = 0.06 mm; (b) Vc = 86 m/min, H = 0.12mm.1004.4. Results and DiscussionHowever, after the tool moves away and the machined surface is immersed within the cuttingfluid, there is an obvious deviation in stress and strain profiles between dry and wet cutting as shownin Fig. 4.22. While the tensile stress exhibits a continuous increase in dry cutting, when the cuttingfluid is applied, the stress either remains constant or displays a slight reduction. At the same time,the plastic strain remains tensile but keeps decreasing, with a higher reduction rate of plastic strainin dry cutting compared to wet cutting. In the cooling down process, the tensile stress and thereduction of the tensile strain originate from the shrinkage of the surface materials, and the finalvalue depends on the deformation of the workpiece material driven by surface thermal gradients.The use of cutting fluid reduces the temperature on the machined surface by carrying the heataway from the workpiece, thereby mitigating the thermal gradients on the machined surface. Toquantify the reduction, the average thermal gradients are calculated based on the simulations, withthe results shown in Fig. 4.23. It is evident that the application of cutting fluid reduces the averagethermal gradients along the horizon and vertical directions by 54.4% and 22.5%, respectively. Thereduced thermal gradients enable a more uniform shrinkage of the surface material during coolingdown. As a result, the magnitude of the surface tensile residual stress after wet cutting is lowercompared to dry cutting; whereas the reduction rate of the tensile plastic strain in the cooling downprocess during wet cutting is less compared to dry cutting.Figure 4.23: Simulated thermal gradients on machined surfaces when orthogonal cutting Was-paloy: (a) Vc = 86 m/min, H = 0.06 mm; (b) Vc = 86 m/min, H = 0.12 mm.Fig. 4.24 compares the machined surface microstructure between dry and wet cutting at differentcutting conditions. It is shown that the white layer occurs on the machined surface after dry cutting,while it disappears when the cutting fluid is applied. As summarized by Ulutan and O¨zel [160], theformation of the white layer is caused by severe plastic deformation, higher temperature gradients,and phase transformation. The above factors are related to the generation of surface residual stressin the metal cutting processes, as Withers and Bhadeshia [172] reported. In addition, the whitelayer also contributes to the generation of surface tensile residual stress. Ko¨nig et al. [84] foundthat the formation of the white layer increases the surface tensile residual stress by about 800 MPa1014.5. Summaryduring turning 16MnCr5 with worn inserts. In this study, the white layer after dry cutting resultsin a higher surface tensile residual stress by around 300 MPa compared to wet cutting. This isowing to higher thermal gradients generated on the machined surface in dry cutting, resulting insevere thermal shrinkage during the cooling down process.Figure 4.24: Surface microstructure after orthogonal cutting Waspaloy with Vc = 86 m/min (a)H = 0.06 mm, dry cutting; (b) H = 0.06 mm, wet cutting; (c) H = 0.12 mm, dry cutting; and (d)H = 0.12 mm, wet cutting.4.5 SummaryIn this chapter, a fully coupled thermo-mechanical model is developed to simulate the orthogonalcutting of Waspaloy with the application of cutting fluid. Based on this model, the effects of cuttingfluid on the local normal/shear contact stress, workpiece temperature, and surface residual stressare quantitatively simulated and discussed, and the formation mechanism of surface residual stressin wet cutting is explained. The following conclusions are drawn:(1) The proposed FEM-based model integrates a two-stage procedure, which first uses the CELtechnique to simulate the tool/workpiece/fluid interaction in the material removal process, followedby the Lagrangian-based model for cooling down procedure to improve the simulation efficiency.The fluid flow state and the velocity distribution close to the cutting tool edge and the machinedsurface are simulated. The comparison between the simulations and the experimental results oncutting forces and surface residual stress profiles demonstrate that the proposed two-stage modellingprocedure is capable of predicting the thermo-mechanical behavior of the workpiece material withthe application of cutting fluid, thus determining the machined surface residual stress.1024.5. Summary(2) Although wet cutting leads to a slight decrease in cutting force, the use of cutting fluid hasno impact on the average local contact stress experienced by the tool edge below the stagnationpoint. The local contact stress is essential in the formation of residual stress caused by mechanicalloads, and wet cutting shares the same residual stress profile as dry cutting if the thermal effect isnot included. However, under the given cutting conditions in this study, all the machined surfacespresent tensile residual stress, whose generation is dominated by thermal loads. The analyses ofplastic strain/stress evolution curves suggest that the final tensile residual stress depends on thethermal shrinkage of the workpiece material during cooling down, and the developed model is ableto capture the tensile residual stress on machined surfaces with acceptable accuracy.(3) The heat generated at the tool edge remains unaffected by cutting fluid, because it is notable to reach the tool-workpiece interface. However, the use of cutting fluid yields a substantialdecrease in surface temperature at around 60 °C. The reduction is attributed to the occurrence offorced heat convection at the fluid-structure interface, which enables the removal of more heat fromthe machined surface compared to dry cutting. As a result, the maximum workpiece temperatureoccurs beneath the machined surface in wet cooling conditions.(4) The use of cutting fluid mitigates thermal gradients between the surface and bulk materialsafter the tool is removed. The reduction in thermal gradients leads to a more even contraction ofthe surface material during cooling down. Therefore, a decrease in surface tensile residual stress isobserved when machining with cutting fluid compared to dry cutting. The high thermal gradientin dry cutting also results in white layer formation at the machined surface in dry cutting, whilethe white layer disappears when the cutting fluid is applied.103Chapter 5Surface Residual Stress in High-SpeedMachining Processes with SerratedChip Formation5.1 Introduction and Literature ReviewIncreasing cutting speeds can enhance the productivity of post-machining AM processes. In high-speed machining of titanium- and nickel-based components, shear localization is a dominant mech-anism with the formation of serrated chips. Compared to continuous chip formation conditions, theoccurrence of shear localization is accompanied by higher cutting temperatures [83], oscillations incutting forces [9], and rapid tool wear progression [68], all of which impose limitations on manufac-turing efficiency and deteriorate the machined surface quality. Analogue to Chapter 4, this chapterexamines the effect of shear localization in chips on the final surface residual stress distribution inthe machining of nickel-based alloys.Experimental investigations in the literature have shown that when serrated chips occur, themachined surface properties vary periodically in the cutting velocity direction. Su et al. [152]noticed periodic undulations of the finished surface profile along the cutting velocity direction,whose frequency is equal to the frequency of shear localization in the chips. In addition, as observedby Xu et al. [175], the thickness of the plastic deformation zone at the finished surface fluctuatedwith the same frequency as that of the shear band formation. The aforementioned variations inmachined surface integrity suggest that thermo-mechanical loads exerted by the cutting tool onthe machined surface are time-varying along the cutting velocity direction during serrated chipformation.Surface residual stress is one of the most fundamental factors of machined surface integrity.The formation of machining-induced residual stress is closely related to the thermo-mechanicalloads applied by the cutting tool on the finished surface during the machining process [159]. Givenits immense technological significance, investigating tool-workpiece interaction is critical to com-prehensively understanding the coupling effects of shear localization in chips and residual stressvariation on machined surfaces.Featuring the machining-induced residual stress profile can be achieved through both experi-mental and modelling approaches. Current residual stress measurement techniques, like XRD andhole-drilling, have limited spatial resolution, which hinders their ability to quantify changes in sur-1045.1. Introduction and Literature Reviewface residual stress caused by shear localization in chips. For example, the size of the X-ray beamtypically exceeds 0.5 mm, whereas, in serrated chip simulations, the average distance between twoneighboring peaks of the surface residual stress curves is around 0.2 mm, as reported by [107].FEM-based numerical models offer an alternative. After experimental verification, these modelscan be used to characterize surface residual stress profiles under various cutting conditions, therebysaving time and energy used for experimental trials in industrial applications. Table 5.1 provides anoverview of the representative studies in the literature and their main findings. However, it showsthat the assessment of residual stress fluctuations associated with the serrated chip formation isvery limited, as demonstrated by the following studies: Liu et al. [107] developed a 2D FEM-basedmodel using the CEL approach to study the effect of chip segmentation on surface residual stressdistribution in orthogonal cutting of IN718. Their simulations showed that residual stress at thefinished surface varies between -500 MPa to 800 MPa, influenced by shear localization in serratedchip formation. Yang et al. [183] extended this FEM-based model for simulating the wet machin-ing of Waspaloy to include the effect of cutting fluid in the Eulerian region. For both dry andwet machining conditions, the simulation results showed periodical fluctuations in surface residualstress along the cutting velocity direction influenced by the serrated chip formation regime. Thecited prior studies did not fully explain the modulation observed by Liu et al. [107], nor did theyaddress the generation of periodically wavy topographies associated with residual stress variationon the machined surface.Table 5.1: Overview of FEM-based models used for the prediction of machining-induced residualstress profiles in orthogonal cutting of high-strength metallic alloys.References Materials RemarksYang and Liu, 2002 [184] AISI 304 Formulating a new stress-based friction modelto simulate the machined surface residual stressChen et al., 2004 [27] Ti6Al4V Featuring the effects of tool flank wear and thecorresponding chip morphologies on the magni-tude and distribution of machined surface resid-ual stressesOuteiro et al., 2006 [125] AISI 316L Predicting the machined surface residual stressafter sequential cutsUmbrello et al., 2010 [162] AISI 52100 Incorporating the microstructural phase trans-formation (including the formation of white anddark layers) for predicting residual stress on themachined surfaceO¨zel and Ulutan, 2012 [129] Ti6Al4V Simulating machining-induced residual stressthrough a temperature-dependent flow-softening material constitutive model1055.2. Model Description and ValidationReferences Categories RemarksKortabarria et al., 2016 [86] IN718 Performing sensitivity analysis to study the in-fluence of material property parameters on themachining-induced residual stress predictionSong et al., 2019 [149] Ti6Al4V Mapping the surface residual stress after themulti-step cutting process with the considera-tion of pre-stress statesLiu et al., 2021 [108] IN718 Investigating the thermo-mechanical loads andresidual stress states under various cutting toolgeometriesXu et al., 2021 [176] Ti6Al4V Modelling chip morphologies and residual stressprofiles in high-speed machining processes con-sidering the in-process state of stressHu et al., 2023 [66] Ti6Al4V Comparing the difference in predicting themachining-induced residual stress profiles undervarious FEM-based formulationsYang et al., 2023 [183] Waspaloy Quantifying the reduction in surface tensileresidual stress attributed to the application ofcutting fluidsTo address the identified gap, in this chapter, we modified the FEM-based machining modeldeveloped in Chapter 4 to focus on investigating the effects of chip morphologies (including con-tinuous and serrated chips) on the surface residual stress distribution in the orthogonal cutting ofWaspaloy, without the application of cutting fluid. Simulations at different cutting speeds real-ize both continuous and serrated chip formations. The accuracy of the simulated chip geometriesand machined surface profiles is validated through comparisons with experimental measurements.Based on the validated models, we compare the time-domain evolution of contact forces and the cu-mulative contact forces below the stagnation point in both continuous and serrated chip formationconditions. The results offer an understanding of the formation of surface periodic wavy featuresand surface residual stress variation considering the generated chip morphologies.5.2 Model Description and Validation5.2.1 Model descriptionFig. 5.1 details the updated FEM-based model to simulate the orthogonal cutting of Waspaloyfor predicting chip morphology, machined surface topography, and surface residual stress distribu-tions. Analogue to the wet machining model presented in Chapter 4, the updated FEM model also1065.2. Model Description and Validationinvolves a two-stage simulation procedure for simulating the final surface residual stress. Initially,in ABAQUS/Explicit mode, an orthogonal cutting submodel built by the CEL technique is usedto simulate material removal and machined surface generation processes. Then, a cooling downsubmodel based on the Lagrangian technique is designed to cool the machined surface to roomtemperature in ABAQUS/Implicit mode. After completing the orthogonal cutting simulation, thenodal temperature, strain, and stress of the Eulerian part selected by the red dashed line box inFig. 5.1 (a) will be transferred into the cooling down submodel presented in Fig. 5.1 (b), to be setas the initial conditions for the Lagrangian nodes.Figure 5.1: Initial geometries and boundary conditions of the updated FEM-based model tosimulate (a) orthogonal cutting and (b) cooling down processes; H refers to the uncut chip thickness,which is set as 0.12 mm in (a).Other key points of the updated FEM-based model are highlighted as follows:1075.2. Model Description and Validation(1) Material constitutive models: The JC plasticity model [74], expressed in Eq. (2.25) and theJC damage initiation model [75], detailed in Eq. (4.1), as well as the Hillerborgâs fracture energymodel [64] are adopted to capture the flow stress and damage behaviors of Waspaloy.(2) Kinematics: In the orthogonal cutting simulation, the workpiece material moves towards thefixed cutting tool with the cutting velocity (Vc) and separates into chips and machined surfaces dueto the tool-workpiece interaction. EVF is employed to track the volume of the workpiece materialoccupied by each Eulerian element during the orthogonal cutting simulation. In the cooling stage,the nodal displacement at the left corner edge (marked as âoâ) of the Lagrangian part is constrainedin both the X and Y directions, whereas the displacement of the Lagrangian nodes at the bottomsurface of the machined workpiece is restricted in the Y direction.(3) Meshing strategies: To describe chip morphology and machined surface stress/strain statesduring orthogonal cutting simulation, the L-shape Euler part to the left edge of the cutting toolor underneath the machined surface is finely meshed by squared elements with an edge length of 2µm. Only one element with a width of 30 µm is assigned to the modelâs transverse (Z) directionand the velocity of the Eulerian nodes has been restricted in the out-of-plane direction. The aimis to develop a 3D model that closely approximates a 2D plane strain configuration. The meshingstrategies in the cooling down submodel are inherited from the orthogonal cutting submodel.(4) Frictional model (only applicable for the orthogonal cutting submodel): The combination ofCoulombâs and shear friction model, as summarized by Zorev [192], is used to include both stickingand sliding behaviors at the tool-chip interface, and the friction coefficient is assigned as 0.5 [183].(5) Heat generation (only applicable for the orthogonal cutting submodel): The fraction of theheat converted from plastic deformation work is set as 0.9. The ratio of the heat generated byfrictional work is fixed at 1.0, with the resultant heat evenly distributed between the cutting tooland the workpiece material.(6) Heat transfer: In the cutting stage, thermal energy is transferred through heat conductionbetween the cutting tool and the workpiece material at the tool-workpiece interface. The thermalconductance is contact-pressure-dependent, and the values are taken from [183]. Heat convectionoccurred between Face a of the Euler part and the environment, with the convection coefficient of10 W/(m2·â) [183]. The heat loss through convection is also performed under the same coefficientbetween Face b (the machined surface) and the environment in the cooling stage. Heat radiationis neglected throughout the simulation.5.2.2 Model validationIn this section, the updated FEM model is validated with experimentally observed chip morphologyand machined surface profiles for two specific cutting conditions, which are labeled as L and Hin Table 5.2. Specifically, Fig. 5.2 compares the simulated chip morphology from the FEM-basedmodel and the measurements based on the turning experiments in [46]. The uncertainties in chipparameters shown in Table 5.3 come from multiple measurements (five times in this study) onat least three chips produced under the same cutting conditions. The same Waspaloy workpiece1085.2. Model Description and Validationmaterial (solution annealed) and cutting tool property (uncoated carbide inserts, with a zero-degree rake angle, a 7-degree clearance angle, and a 20-µm tool edge radius) as [183] are used inthe experiments, and all the tests are performed without the application of cutting fluid. Theobtained results suggest that as the cutting speed increases from 30 m/min ( L ) to 120 m/min( H ), the chip morphologies change from macroscopically continuous ribbons (homogeneous plasticdeformation) to serrated appearances (shear localization). This result aligns with the findings of Suet al. [152], who reported material failure in the primary shear zone due to higher temperatures atincreased cutting velocities, leading to the transition from continuous to serrated chip morphology.Table 5.2: Orthogonal cutting conditions for model validations.Conditions Cutting velocity Uncut chip thickness Width of cutL 30 m/min 120 µm 1.5 mmH 120 m/min 120 µm 1.5 mmFigure 5.2: Comparisons between measured and predicted chip morphologies for cutting condi-tions L and H .1095.2. Model Description and ValidationTable 5.3: Comparison of chip geometries between simulations and experimental measurementsin [46] for cutting conditions L and H .Conditions Parameters Experiments PredictionsL hchip [µm] 276± 10 263± 5 (-4.7%)H h1 [µm] 228± 8 218± 1 (-4.4%)h2 [µm] 160± 6 175± 8 (+9.4%)epitch [µm] 90± 7 77± 12 (-14.4%)Îsawâtooth [°] 57± 6 54± 2 (-5.3%)For a quantitative comparison, Table 5.3 lists the geometric characteristics of the machined chipsfrom simulations and the experimental measurements by Fazlali et al. [46]. Average chip thicknesshchip is used to characterize the continuous chips. According to Agmell et al. [2] and Jomaa et al.[76], the serrated chip morphologies can be featured by four parameters, including the maximumchip thickness h1, the minimum chip thickness h2, the pitch distance epitch, and the saw-toothincluded angle Îsawâtooth. Overall, a discrepancy of less than 15% is observed when compared withexperimental measurements, validating the predictive accuracy of the updated model. It shouldalso be noted that the vertices of the simulated serrated chips are more rounded compared to theexperimentally observed chips. As discussed in [8, 108, 176], this phenomenon is common in theprediction of serrated chip formation using the CEL method, where the workpiece is modelled bythe Eulerian approach. This can be attributed to the accumulation of reconstruction errors at theEulerian material interface. According to [63], in Eulerian analysis, materials are allowed to flowthrough fixed Eulerian elements without deformation throughout the simulation. Volume fractiondata for each Eulerian material in an element are calculated, and within each time increment,the boundaries of each material are reconstructed using this data. The reconstruction algorithmapproximates material boundaries within an element as simple planar facets. This approach resultsin a simple, approximate material interface, which will lose its original shape after several timeincrements. Therefore, accurately determining a materialâs boundary within an Eulerian elementis feasible only for simple geometries, necessitating fine grid resolution for complex features.Further, a Mitutoyo SURFTEST SV-500 profilometer was used to measure the machined sur-face profiles along the cutting velocity direction. Comparisons between the simulated and measuredsurface profiles for cutting conditions L and H are shown in Fig. 5.3. Both simulated and mea-sured surface profiles change from relatively flat to noticeable wavy profiles from cutting conditionsL and H . In case H , the fluctuation frequency of the surface micro-waves, defined as the ratiobetween the cutting velocity (Vc) and the wavelength of the surface profiles waves (λwave), exhibitsgood agreement between simulations (around 18.2 kHz) and experiments (about 19.3 kHz). Therelation of this periodicity to chip shear localization and its influence on machined surface residualstress will be highlighted in the following section.1105.3. Results and DiscussionsFigure 5.3: Measured and simulated surface profiles along the cutting velocity direction for cuttingconditions (a) L and (b) H .Overall, the above results demonstrate that the modified FEM-based model and the method-ology employed effectively capture the mechanics of both continuous and serrated chip formation.This approach enables a detailed examination of the underlying physics in the machining process,which will be discussed in the next section.5.3 Results and Discussions5.3.1 Tool-workpiece contact forcesThis section compares the distribution, directionality, and evolution of contact forces at the tool-workpiece interface between two cutting conditions L and H . As shown in Fig. 5.4 and 5.5,contact forces per unit width in the normal (CNORMF) and shear (CSHEARF) directions aredepicted through vector plots, wherein the relative magnitudes are indicated by the length andcolor of the vectors. Here, the values of CNORMF and CSHEARF are determined based on thecontact forces reported by the tool element at its interface with the workpiece.1115.3. Results and DiscussionsFigure 5.4: Distribution and evolution of contact forces at the tool-workpiece interface duringcontinuous chip formation (case L ): Contact forces distribution in (a) normal and (b) sheardirections at t = 0.43 ms; Contact forces evolution at (c) Point A, (d) Point S, and (e) Point B.Figure 5.5: Distribution and evolution of contact forces at the tool-workpiece interface duringserrated chip formation (case H ): Contact forces distribution in (a) normal and (b) shear direc-tions at t = 0.43 ms; Contact forces evolution at (c) Point A, (d) Point S, and (e) Point B.1125.3. Results and DiscussionsTo better understand the evolution of contact forces at the tool-workpiece interface, threepoints A, S, and B are chosen in Fig. 5.4 and 5.5, and the time-domain variations of CNORMFand CSHEARF are recorded. Point A is located on the toolâs rake face, at a distance of 45 µmabove the intersection point between the round edge and the rake face of the cutting tool. Thenominal stagnation point is identified as Point S, where the incoming workpiece material inherentlychanges directions, indicating the necessity of a stagnation point on the cutting tool. The materiallocated above this point is pushed towards the toolâs rake face, contributing to chip formation,while material below this point is plowed underneath the cutting tool, resulting in the machinedsurface. The stagnation point can also be understood through the direction of CSHEARF, as thematerial flow direction on the toolâs round edge aligns with CSHEARFâs direction. Finally, PointB is defined as the location on the tool flank face to record the time-varying contact forces. Thispoint is positioned within the contact region between the tool flank face and the newly machinedsurface, specifically at the last contact point with the machined surface in case L .For cutting condition L , Fig. 5.4 (a) and (b) represent a uniform CNORMF in the centralportion of the tool-chip interface and a monotonically increasing CSHEARF moving away fromthe round edge to the rake face, respectively. The time evolution of CNORMF and CSHEARFare shown in Fig. 5.4 (c), (d), and (e). The results show that both CNORMF and CSHEARFstabilize quickly at Points A, S and B. Furthermore, the stagnation point is stationary during theorthogonal cutting simulation, implying a constant ratio between the volume of material passingthe shear plane to form the chip and that of the material plowing by the cutting tool to generatethe machined surface.However, for cutting condition H in Fig. 5.5, the results suggest that shear localization resultsin distinct profiles in CNORMF and CSHEARF. Fig. 5.5 (a) and (b) demonstrate that at the tool-chip interface, normal and shear contact forces reach the peak values when the shear band contactsthe tool rake face between each chip segment. In addition, Fig. 5.5 (c), (d), and (e) show that atPoints A, S and B, fluctuations occur with different characteristics. For Point A, the advancementof the tool and the generation of a new chip segment cause the previously formed shear bands tomove upwards along the rake face. As the shear band passes over A, the highly concentrated normalstress and shear strain lead to CNORMF and CSHEARF reaching their peak values, followed bya decrease after the shear band moves away. The examination of the contact forces at Point Sindicates that, unlike case L , the location of the stagnation point on the cutting tool movescyclically during serrated chip formation. As a result, Point S aligns with the stagnation pointonly when CNORMF reaches its maximum, while CSHEARF approaches zero. To investigate thisphenomenon in more detail, Fig. 5.6 presents the distribution of shear contact forces for cases Land H at various time instants within the formation of one chip segment. The stagnation angle(θs) denotes the angular location of the stagnation point on the cutting tool. It can be seen that forcase H , at the onset of chip segment generation (t = 0.43 ms), the stagnation point is located nearwhere the flank face contacts the machined surface. During the chip segment formation, this pointgradually moves toward the toolâs rake face. Once the chip segment generation is finalized and a1135.3. Results and Discussionsnew shear band forms, the stagnation point stops moving further on the cutting tool. Finally, att = 0.48 ms, to accommodate the formation of the subsequent chip segment, the stagnation pointshifts back to the contacting point on the flank face side.Figure 5.6: Effects of chip morphologies on the distribution of shear contact forces and stagnationangles.Meanwhile, as illustrated in Fig, 5.5(e), CNORMF and CSHEARF recorded at Point B peri-odically vary under case H . To better understand the time-domain evolution of contact forces atPoint B for case H , the time-varying contact forces at Point B for case L need to be explained. Incase L , continuous chips are produced. This results in the generation of a fixed stagnation point,leading to a constant volume of material being plowed by the cutting tool. Therefore, the amountof elastic recovery remains unchanged throughout the cutting process, resulting in a constant con-tact length between the tool flank face and the newly generated surface. As Point B is positionedat the last point where the tool flank face contacts the newly machined surface, CNORMF andCSHEARF are almost zero in case L . In case H , the variation of contact forces at Point Bcan be attributed to the change of contact length on the tool flank face, which is associated withthe variation in the amount of elastic recovery of the workpiece material after passing the toolâslowermost point. Zhuang et al. [191] have found that the amount of elastic recovery increases withthe volume of material being plowed. In one chip segment formation period, once the stagnation1145.3. Results and Discussionsangle is greater than the fixed stagnation angle in case L , the volume of workpiece material beingplowed by the cutting tool increases. This results in an increase in the amount of elastic recoveryof the plowed material. As a result, the contact length between the tool flank face and the newlygenerated surface increases. Therefore, the contact forces at Point B start to increase from zero.CNORMF and CSHEARF at Point B will reach their peaks once the stagnation angle shifts to itslargest value. After that, CNORMF and CSHEARF at Point B decrease as the stagnation angledecreases. Once the stagnation angle is smaller than the fixed stagnation angle in case L , PointB on the tool flank face will no longer be in contact with the newly generated surface. As a result,CNORMF and CSHEARF at Point B become zero.5.3.2 Machined surface topographyThis section examines the correlation between the simulated machined surface profiles and twoprocess parameters: the stagnation angle evolution and plowed material elastic recovery. As shownin Fig. 5.7(a), for case L , a relatively flat machined surface without any noticeable waviness isobserved. On the other hand, in Fig. 5.7(b) for case H , periodic micro-waves are noticeable withan amplitude of around 0.9 µm and wavelength of about 110 µm.Figure 5.7: Comparisons of machined surface topographies at cutting conditions (a) L and (b)H when orthogonal cutting of Waspaloy.With a closer examination of the stagnation angle θs in Fig. 5.8(a), it is shown that case Lmaintains a constant value of 42.4°, while case H shows a variation from 21.2° to 90°. Examplesat multiple time instants are shown in Fig. 5.8(b). It can be found that when continuous chipsare formed, due to the generation of a constant stagnation angle in the orthogonal cutting process,the volume of workpiece material plowed by the cutting tool is constant. This results in a uniform1155.3. Results and Discussionsamount of elastic recovery of the plowed workpiece material after the tool moves away. As a result,a flat machined surface is obtained in the simulation. Conversely, as shown in Fig. 5.8(b), thegeneration of a chip segment involves the time-varying stagnation angle in the cutting process.During one chip segment formation period, a larger stagnation angle results in more workpiecematerial being plowed by the cutting tool. Consequently, there is a larger amount of elastic recoveryof the plowed workpiece. Therefore, periodically wavy features are formed on the machined surfacewith the movement of the cutting tool.Figure 5.8: Relationship between stagnation angles and machined surface topographies at cuttingconditions L and H : (a) Evolution curves of stagnation angle; (b) Detailed description of thegeneration of machined surface topographies.5.3.3 Machined surface residual stress and plastic strainThe effect of the time-varying location of the stagnation points on the surface residual stress andthe equivalent plastic strain is discussed in this section. Only machined surface residual stress inthe cutting velocity direction (Ïrexx) is studied, due to its key role in the fatigue life of machinedcomponents. Fig. 5.9 (a) and (b) show Ïrexx distribution for cases L and H , respectively. For case1165.3. Results and DiscussionsL , residual stress gradients can be noticed only in the uncut chip thickness direction. However,case H shows residual stress gradients in both cutting velocity and uncut chip thickness directions.Quantitative results of Ïrexx at the machined surface are presented in Fig. 5.9 (c). For case L , aconstant tensile residual stress of 1,298 MPa is observed at the machined surface, while case Hshows the fluctuations of the residual stress with a peak-to-valley amplitude of around 160 MPa.Figure 5.9: Surface residual stress distribution at cutting conditions (a) L and (b) H ; (c)Comparison of residual stress profiles on the machined surfaces between cutting conditions L andH .Variation of Ïrexx can be related to the time-varying thermo-mechanical loads that the cuttingtool acted on the machined surface during the machining process. To explain the Ïrexx variationshown in Fig. 5.9 (c), the total contact forces below the stagnation point are studied. Fig. 5.10(a) and (b) illustrate the CNORMF distribution on the cutting tool under both cutting conditionsL and H at t = 0.455 ms, respectively. Fig. 5.10 (c) and (d) present the evolution of thesum of CNORMF and CSHEARF below the stagnation point for cutting conditions L and H ,respectively. For case L , CNORMF and CSHEARF sums below the stagnation point remainunchanged throughout the cutting process, with an average value of around 28.2 N/mm and 4.5N/mm, respectively. This is due to the generation of a fixed stagnation point associated with thecontinuous chip formation, leading to a constant volume of material compressed by the cuttingtool. As a result, a constant thermo-mechanical load is involved in the residual stress formation,generating a uniform residual stress distribution on the machined surface. However, in case H ,the cyclical motion of the stagnation point results in variations in local contact forces. Specifically,at the beginning of a chip segment formation, the stagnation point is located near the toolâs flankface, corresponding to the least amount of material plowed. As a result, the local CNORMF andCSHEARF sums reach the minimum values of about 14.8 N/mm and 2.7 N/mm, respectively.Then, as the chip segment formation proceeds, the stagnation point continuously shifts toward thetoolâs rake face side, resulting in a larger amount of workpiece materials plowed by the cutting tool,1175.3. Results and Discussionstherefore generating an increase in the total contact forces acting on the cutting tool. At the end ofa chip segment formation period, CNORMF and CSHEARF sums reach their maximum values, ataround 87.5 N/mm and 5.1 N/mm, respectively. Consequently, changes in the sum of the contactforces below the stagnation point cause fluctuations on the machined surface residual stress. Thesefluctuations occur at the same frequency as the cyclic motion of the stagnation point, which, aspreviously mentioned, coincides with the chip segmentation frequency. This analysis reveals thecoupling effect between the periodic chip segmentation and the spatial variation of the residualstress on the machined surface. The higher average CNORMF and lower average CSHEARF incase H as opposed to case L also explain the lower tensile (more compressive) surface residualstress in case H shown in Fig. 5.9 (c). Thus, the correlation between the periodicity of chipsegmentation and spatial variation in surface residual stress is clearly established.Figure 5.10: The distribution of normal contact stresses on the cutting tool when t is 0.455 msunder the cutting conditions (a) L and (b) H ; The evolution of the sum of (c) CNORMF and (d)CSHEARF below the stagnation point during simulations under the above two cutting conditions.Gradients in equivalent plastic strain are useful to determine the machined surface metallo-graphic presentation. Fig. 5.11 (a) and (b) show the distribution of the equivalent plastic strainat the machined surface under cutting conditions L and H , respectively. For case L , gradientsin the plastic strain can only be seen in the uncut chip thickness direction. However, for case H ,the concentrated plastic strain is periodically distributed on the machined surface in the cuttingvelocity direction. A quantitative examination of the plastic strain at the machined surface in Fig.5.11 (c) shows a uniform value of around 3.76 for case L . However, for case H , a fluctuatingvalue, ranging between 2.05 and 4.73, can be observed, with the same frequency as the chip seg-1185.4. Summarymentation. The former uniform value is associated with a constant material volume plowed, whilethe fluctuating strain in the latter is due to the varying plowed material volume caused by thecyclical motion of the stagnation point, as described before.Figure 5.11: Distribution of the equivalent plastic strain under the cutting conditions (a) L and(b) H ; (c) Comparison of the equivalent plastic strain curves at the finished surfaces under theabove two cutting conditions.5.4 SummaryThis chapter investigates the effect of shear localization in the chip formation region on the sur-face residual stress distribution in the orthogonal cutting of Waspaloy. Based on the FEM-basedsimulations of the cutting process, the analyses of the normal/shear contact forces at the tool-workpiece interface demonstrate that the stagnation point remains stationary on the cutting toolduring continuous chip formation without shear localization. This results in a machined surfacewithout noticeable waviness and is characterized by a uniformly distributed residual stress on thefinished surface. However, at a higher cutting velocity, the occurrence of serrated chips due tothe periodic shear band formation causes the stagnation point to cyclically move on the cuttingtool. This movement results in varying volumes of workpiece material being plowed by the cuttingtool, generating waviness at the finished surface due to periodicity in the elastic recovery of plowedmaterials. Furthermore, this cyclic motion induces time-varying thermo-mechanical loads betweenthe tool and the machined surface. Therefore, residual stress at the machined surface fluctuatesalong the cutting velocity direction with the same frequency as chip segmentation.119Chapter 6Surface Residual Stress FollowingMachining of Directed EnergyDeposition Structures6.1 Introduction and Literature ReviewDED-fabricated components are characterized by high-gradient bulk residual stress (due to rapidheating and cooling cycles experienced by the material) [167], inhomogeneous grain size (caused bynon-uniform solidification rates) [106], as well as rough surface finishes (attributed to inconsistentmelt pool dynamics and incomplete melting of powder particles) [127]. Therefore, effective post-processing strategies, like heat treatments and machining, are essential to maximizing the potentialof DED-produced structures [77]. For example, post-deposition heat treatments involve exposingthe deposited material in a controlled temperature environment, to refine grain size [186], enhancemechanical properties [140], and release bulk residual stress [73]. Moreover, post-machining DEDprocesses aim to improve surface finish, ensure dimensional accuracy and tolerance, as well as allowcustomization for specific application requirements [114].When machining metallic components, non-uniform plastic deformation of workpiece materialat the tertiary zone leaves residual stress on the finished surface [183]. Apart from the residual stresscaused by phase transformation, the final residual stress relies on the accumulated plastic stressinduced by thermo-mechanical loads exerted by the cutting tool on finished surfaces [172]. Servingas one of the most fundamental factors in surface integrity, surface residual stress plays an importantrole in the corrosion resistance and fatigue life of machined components [170]. Therefore, criticalassessment of the final surface residual stress after post-machining of DED-created structures issignificant.Experimental investigations have demonstrated that the machinability of DED-fabricated sam-ples is influenced by both the initial residual stress states and microstructure distributions. Onthe one hand, DED-induced bulk residual stress affects both chip morphologies and final surfaceresidual stress states. For instance, Lane et al. [89] observed that chip bending occurred in thedirection perpendicular to the transverse (width of cut) direction when orthogonal cutting of AM-produced 17-4 PH stainless steel. They attributed these sideway chip curls to the stress relaxationof removed materials after plastic deformation. Du et al. [43] conducted side milling experimentson Ti6Al4V blocks fabricated by DED. Based on XRD measurements, they quantified the differ-1206.1. Introduction and Literature Reviewences in residual stress profiles before and after post-milling. Their results suggest that subsequentmilling operations effectively mitigated the original tensile residual stress within DED parts, withthe reduction magnitude closely related to the radial depth of cut. Specifically, when a smallerradial depth of cut (0.1 mm) was employed, residual stresses on the subsequently milled surfaceremained in a tensile state but lower than the initial value. As the radial depth of cut increasesfrom 0.1 mm to 0.3 mm, the final residual stress on the post-milled surfaces underwent a gradualtransition from tensile to compressive. On the other hand, the inhomogeneity in DED-producedgrains has a notable influence on their machinability. Xue et al. [177] performed post-milling testsof CoCrFeNiMn high entropy alloy (HEA) produced by casting and DED, respectively. The ob-tained results showed that compared to the conventionally cast HEA, higher cutting forces and toolwear rates were generated when post-milling DED-created samples. The same trend can also be ob-served in [15], wherein examinations in cutting forces and tool wear were conducted and comparedduring the machining of ASTM A131 steel fabricated by hot-rolling and DED processes. Moreimportantly, due to the spatially non-uniform solidification rates associated with DED processes,the resulting microstructures present heterogeneity within the DED sample. As concluded by Koket al. [82], the divergences in microstructure led to anisotropy in both mechanical properties andfunctional performances of DED-fabricated components. As a result, thermo-mechanical responesof DED-produced samples were direction-dependent during post-machining. Bai et al. [15] per-formed milling tests of ASTM A131 steel produced by DED. The obtained results indicate thatcutting forces in the milling of DED-top samples (featured by strip melt tracks and cellular grains)were always lower than the forces generated in the milling of DED-side samples (characterized byarc and strip melt tracks with dendritic grains). Given that the resultant cutting forces can bean effective indicator for estimating the thermo-mechanical loads acted on the finished surface, itis essential to consider the initial distribution of DED-generated microstructure in predicting thepost-machined surface residual stress. The above analyses suggest that, to achieve satisfactory sur-face residual stress states after post-machining, it is imperative to quantify how the DED-inducedinitial residual stress and microstructure distribution determine the final residual stress.Owing to their cost-effectiveness and versatility, FEM-based numerical models have been exten-sively used to simulate metal cutting processes [11, 106]. Outeiro et al. [126] critically evaluated theeffectiveness of present FEM-based models in predicting machining-induced residual stress. Build-ing on these studies, recent FEM models specifically developed for post-machining simulations haveexamined how the DED-produced bulk residual stress affects the final surface residual stress. Forinstance, Sunny et al. [154] performed end-milling simulations of DED-fabricated IN625 thin-wallstructures. Their findings revealed the significant effects of the DED-induced bulk residual stresson the final surface residual stress after post-machining. These effects varied greatly under thegiven two post-machining strategies, i.e., horizontal- and vertical-oriented tool paths. Specifically,when employing horizontal tool paths, the final surface residual stress along the feed directionranged from -180 MPa to 80 MPa with the considerations of DED-produced residual stress, andfrom -20 MPa to 200 MPa without considering it. Likewise, under vertical tool paths, the final1216.1. Introduction and Literature Reviewsurface residual stress perpendicular to the feed direction varied from -100 MPa to 100 MPa withthe consideration of DED-produced residual stress, and from -150 MPa to 50 MPa without consid-ering it. In addition, Luo et al. [112] also built up a 3D FEM model to predict the final surfaceresidual stress after post-milling of DED-fabricated SS316L stainless steel blocks considering theinitial residual stress. To decouple the joint effects between DED-induced residual stress and themilling-produced residual stress, they separated the final residual stress profiles along the depthdirection into three regions: the work hardening layer, the elastic recovery zone, and the elasticstability zone. The simulations suggested that within the work hardening layer, the final residualstress mirrored the milling-induced residual stress profiles; while in the elastic recovery zone, it fol-lowed a linear superposition of DED- and milling-induced residual stress profiles; eventually, due tothe recovery of milling-induced residual stress to zero, the final residual stress in the elastic stabil-ity zone matched the originally DED-induced residual stress profiles. However, the above findingshave not yet been experimentally validated. Moreover, in a recent work done by Lee et al. [90],a comprehensive modelling procedure using FEM was developed to simulate fully-coupled additiveand subtractive manufacturing operations. This model was designed to capture the evolution ofresidual strain/stress in the fabrication of SS316L thin-wall structures (comprising five sections)through hybrid DED + milling processes. The modelâs effectiveness was quantitively assessed bycomparing the simulated residual strain profiles with experimental measurements obtained by neu-tron diffraction techniques. Results demonstrated that after fabricating the fifth section of the thinwall using DED, subsequent top-surface milling operations transformed the top surfaceâs residualstress from weakly tensile to highly compressive states. To counterbalance the resultant compressiveresidual stress induced by top-surface milling, tensile stress accumulated at the interfaces betweeneach section as cutting progressed towards the wallâs side surfaces. However, one limitation of thiswork is that their model neglects the surface residual stress generation in the subsequent machiningoperations. This is because, in their post-machining simulations, material removal is modelled byelement deactivation based on the model change method. This method allows for deleting elementsaccording to a predefined tool path, without considering any chip formation and surface generationmechanics caused by tool-workpiece interactions during actual cutting processes. As a result, nothermo-mechanical loads are applied on the newly generated surface to form milling-induced resid-ual stress. Therefore, their simulations only quantify the redistribution of the bulk residual stressdue to the rigid removal of a specific volume of material. It should also be emphasized that all theaforementioned approaches ignore the microstructure effects.As highlighted by Sunny et al. [153], in numerical models, considering microstructural featuresis essential for accurately simulating material responses in the post-machining of AM-generatedmetallic components. Careri et al. [24] developed a 3D FEM model to predict the cutting forces andtemperature in the radial turning of DED-fabricated IN718 cylinders. To feature the anisotropic be-havior of the as-deposited material, they adopted a microstructure-based material constitutive lawproposed by Yuan et al. [187]. This material constitutive law characterizes the flow stress of DED-fabricated IN718 by incorporating three stress components, including the equivalent anisotropy1226.2. Post-Machining DED Model Descriptioncomponent (obtained by analyzing the orientation of the initial grain structure), the athermalcomponent, and the equivalent thermally activated component. The simulated cutting forces andtemperature were verified through experimental measurements. Later, in [25], they extended theirFEM model to calculate the surface residual stress profile after radial turning operations. However,the model exhibited limited accuracy in predicting the final residual stress, particularly on thepost-turned surface. In addition to their modelling limitations discussed in [25], the neglection ofinitial residual stress could also explain the stress deviation between predictions and measurements,as the authors in [25] assumed that the as-deposited workpiece is stress-free during post-machining.This neglection would lead to substantial discrepancies between simulated and measured residualstress values, as the initial residual stress states play a critical role in shaping the final residualstress profiles, as highlighted in [154].The literature above demonstrates that in the prediction of the final residual stress followingmachining of DED components, the existing FEM-based models usually operate under the assump-tion that the workpiece is either stress-free or that microstructural effects are negligible. However,how the DED-induced bulk residual stress or microstructural features contribute to the generationof final residual stress after machining has not been systematically understood. Meanwhile, thecumulative interactions between DED-induced initial bulk residual stress and microstructure onthe post-machined surface residual stress remain largely unexamined, which are seemingly essentialfor forecasting part behaviour post repair. To address the identified research gaps, in this thesis,the FEM-based model presented in Chapter 4 is extended to incorporate both the initial residualstress states and microstructure characteristics, to describe the flow stress of DED-deposited ma-terials. This framework, presented in this chapter, focuses on predicting the final surface residualstress after the orthogonal cutting of DED-built single-track IN718 samples, which can also beapplied to other post-processing techniques of any geometric features produced by DED or otherAM-produced metallic components. The developed model achieves the following objectives: (1)capture the penetration depth of post-machining into the initial residual stress profile, (2) revealthe formation mechanism of final surface residual stress in the post-machining of DED processes,and (3) quantify how the initial residual stress states and grain size values determine cutting forcesand final surface residual stress profiles. The mechanisms uncovered can guide the selection ofany existing post-treatment strategies to engineer the final surface residual stress, especially forDED-repaired metallic components.6.2 Post-Machining DED Model DescriptionFig. 6.1 details the FEM-based numerical modelling framework by taking an orthogonal cuttingsimulation of a DED-fabricated thin-wall structure as an example.1236.2. Post-Machining DED Model DescriptionFigure 6.1: A schematic diagram of post-machining DED simulation showing the interactionbetween consecutive processes and their respective stress conditions.As illustrated in Fig. 6.1, two critical factorsâinitial residual stress states and microstructuralcharacteristicsâare integrated to feature the thermo-mechanical response of DED-created materi-als during post-machining processes. The initial stress conditions within a DED component can beexperimentally measured using XRD or other techniques. Given that the existing residual stressmeasurement techniques typically provide spatially limited data, a residual stress reconstructionalgorithm developed by Sunny et al. [154] is used to generate a complete residual stress map of theentire component. After that, a user-subroutine SIGINI [63] is employed to map the reconstructedresidual stress data into the post-machining DED model. Additionally, the actual grain distribu-tion within a DED component can be obtained by analyzing the EBSD data. Considering thatcolumnar grains are predominant in DED-produced metallic components, the DED-created grainstructures are simplified as ellipsoids to calculate the average grain size in the cutting velocity (x)direction (dg x). A new material constitutive law (including both plasticity and damage models)1246.2. Post-Machining DED Model Descriptionis developed to capture the materialsâ flow stress during post-machining simulations. This consti-tutive law quantifies the effects of initial stress states on materialsâ thermal-mechanical behaviorby the state of stress parameters, including stress triaxiality (η) and Lode parameter (θ¯), and thegrain size indicator dg x, which is used to feature how the actual grain distribution influences thematerialsâ initial yield strength. The developed material constitutive law is incorporated into thepost-machining DED model through user-subroutines VUHARD and VUSDFLD [63].In addition to the details mentioned above, the subsequent sections will cover the material, heatexchange, and tribology input models, along with the geometries of the objects and their meshingstrategies.6.2.1 Initial geometries, meshing strategies, and kinematicsAs shown in Fig. 6.2, to predict the finished surface residual stress profile after the orthogonalcutting of DED-built single-track IN718 samples, three steps are required: initial residual stressreconstruction (Step 1), orthogonal cutting (Step 2), and cooling down (Step 3). Using XRDtechniques, the measured residual stress values represent the average values within the regionirradiated by the X-ray beam. Therefore, we neglect the variation of actual residual stresses in bothx and y directions and their influences on the final residual stress. Meanwhile, as demonstratedin [183], the FEM models based on the CEL method can only be implemented in 3D. As a result,to ensure that the updated 3D CEL-based orthogonal cutting model approximates a 2D modelunder the plane strain condition, we only employed one element in the transverse (y) direction forsimplification, whose length is equal to the average grain size of the single-track IN718 samplesin the transverse (y) direction. The calculation of dg y will be discussed in Section 6.3.4. Furtherdetails of the FEM-based models in each step are outlined below:(i) Step 1 is performed using the Lagrangian method under ABAQUS/Implicit mode. Theaim is to create a complete residual stress map in the âfocal areaâ (see Fig. 6.2) based on a limitednumber of residual stress measurement data acquired by XRD. As demonstrated in Fig. 6.2, theâfocal areaâ, also referred to as the âresidual stress reconstruction areaâ, shares the same length of1 mm as the X-ray beam diameter in the x direction. Meanwhile, its length along the z-directionis set to 0.35 mm (see Fig. 6.2). This corresponds to the location where post-machining-inducedresidual stress profiles start to follow the initial residual stress profile, which will be discussed inSection 6.3.5. More importantly, in the residual stress reconstruction model, the âfocal areaâ issurrounded by an âexpanded areaâ on three sides, with a 0.5 mm extension from each side in thex and/or z directions. This extension ensures that stress equilibrium and compatibility can bemaintained, preventing artificial boundary effects (the defined displacement constraints on eachside of the FEM model), to rebuild a more realistic residual stress distribution. A total numberof 68,000 8-node Lagrangian brick elements (C3D8R) are employed in this analysis, and each ofthe elements shares a uniform length of 5 µm, which corresponds to a ratio of around 1:10 relativeto the actual grain size along the x direction. The reconstructed results are presented in Section6.3.3. Once Step 1 is completed, the reconstructed residual stress fields will be assigned as the1256.2. Post-Machining DED Model Descriptioninitial stress conditions for the orthogonal cutting model in Step 2.(ii) Step 2 captures the thermo-mechanical response of DED-built IN718 samples during or-thogonal cutting processes, with the consideration of the initial residual stress and grain size ef-fects. Under the ABAQUS/Explicit mode, the orthogonal cutting model is established by the CELmethod, where the cutting tool is formulated by the Lagrangian approach, and the workpiece ismodelled by the Eulerian method. As shown in Fig. 6.2, at the beginning of the post-machiningsimulation, the Euler domain marked as blue, which shares the same dimensions as the Lagrangianpart in Step 1, is occupied by the workpiece material (IN718). Meanwhile, the remaining areas ofthe Euler part are void. To increase the prediction accuracy for the surface generation, the Eulerpart ahead of the tool edge is more finely meshed into squared elements, with an edge size of 5µm (the same as Step 1). There are 4,285 8-node thermally coupled Lagrangian brick elements(C3D8RT) and 57,486 8-node thermally coupled linear Eulerian brick elements (EC3D8RT) in thisthermo-mechanical analysis. As the tool advances, EVF is employed to monitor the volume of thematerial occupied by each Eulerian element. Other boundary conditions of the orthogonal cuttingmodel are shown in Fig. 6.2. In addition, the friction coefficient between the cutting tool and theworkpiece material (IN718) is defined as a function of cutting velocity (Vc), as reported in [14].The fraction of friction-generation heat is set as 1.0 [183], and the generated heat is evenly dividedbetween the cutting tool and the workpiece. Thermal conductance between the cutting tool andthe workpiece is contact-pressure-dependent, with the conductance coefficients adopted from [1].During the orthogonal cutting simulation, heat convection occurs between the machined workpiece(referred to as the âFace aâ in the Eulerian region, shown in Fig. 6.2) and the environment, and theheat convection coefficient is assigned as 10 W/(m2·â) [183]. Considering that the total simulationperiod is 3 ms, heat loss caused by radiation is neglected throughout the simulation.(iii) Step 3 is designed to unlock the thermal constraints within the Eulerian elements belowthe finished surface obtained in Step 2. This is achieved in ABAQUS/Implicit mode, as reported in[183], by first transferring all the resulting data (temperature, strain, and stress, obtained in Step2) at each integration point of the Eulerian elements below the finished surface into a Largrangian-based FEM model, followed by cooling the post-machined workpiece to room temperature. Duringcooling down (Step 3), heat convection occurs between the post-machined surface and the envi-ronment, which shares the same heat convection coefficient as Step 2. Once the finished workpiececools to room temperature, the remaining stress state represents the final surface residual stress.1266.2. Post-Machining DED Model DescriptionFigure 6.2: Initial geometry and boundary conditions of the FEM-based numerical modellingprocedure to predict the final residual stress after post-machining of DED-built single-track IN718samples.6.2.2 Material modelsIn the post-machining simulations, the uncoated carbide insert is treated as a rigid body, neglectingelastoplastic deformation from tool-workpiece interaction and any geometric changes caused byinitial and progressive tool wear. The mechanical and thermal properties of the cutting tool aresourced from [183]. Meanwhile, the physical properties of the workpiece material (IN718) aredefined as temperature-dependent, which are adopted from [165]. The material constitutive laws(both plasticity and damage models) are critical in accurately capturing the elastoplastic behaviorof IN718 during machining simulations. The material constitutive laws employed in the post-machining DED simulation are described below.1276.2. Post-Machining DED Model Description6.2.2.1 Material plasticity modelsThe existing material plasticity models used for metal cutting simulation can be divided into twocategories: phenomenological and physics-based models. Phenomenological models are widely em-ployed in metal cutting simulation, due to their simplicity and reduced computational demand[155]. They generally use an empirical equation to describe the macroscopic flow stress as a func-tion of strain, strain rates, and temperature changes, without considering the microscopic detailsof the material [106]. On the other hand, physics-based models include specific microstructuraldevelopment during machining, such as dislocation density, recrystallization, and grain size evolu-tion [37]. These models are mathematically complex and are designed to link the aforementionedmicrostructural changes directly to the macroscopic flow stress of the material [134]. Therefore, theuse of physics-based models requires more computational resources. Herein, we select phenomeno-logical models over physics-based models due to their capabilities in characterizing the cumulativeeffects of the initial stress states and grain size on the materialâs flow stress.In metal cutting simulation, the most commonly used phenomenological material plasticitymodel is the JC plasticity model [74]. It models the equivalent von Mises flow stress (ÏÂ¯JC) by threeterms, as described in Eq. (2.25), including the equivalent plastic strain (ε¯p), the equivalent plasticstrain rate ( ËÂ¯Îµp), and the temperature (T ), respectively. Table 6.1 lists the JC plasticity coefficientsused in this study. Five coefficients in the JC plasticity model including AJC , BJC , CJC , mJC ,and nJC , are taken from [14], where orthogonal cutting tests of IN718 produced by DED wereconducted to calibrate these coefficients.Table 6.1: Coefficients of the JC plasticity model for DED-built IN718 [14, 50, 165].AJC BJC CJC mJC nJC ËÂ¯Îµ0 Tm T0833 MPa 1642 MPa 0.02 1.3 0.244 1 sâ1 1260 °C 20 °CAlthough the JC plasticity model is easy to calibrate, it presents several limitations in providingaccurate simulation results. Firstly, the predicted flow stresses at large strains are incorrect becausethe JC plasticity model allows the material to flow at an indefinite value. Secondly, the JC plasticitymodel updates the materialâs flow stress starting from a fixed yield strength, without accountingfor the change of yield strength due to grain size strengthening effects. Thirdly, any alterations inmicrostructure and their influences on the materialâs flow stress are not included. Finally, the JCplasticity model cannot capture the effects of both the initial and in-process stress states on the ma-terialâs response. Many scholars have modified the original JC plasticity model by including specialadd-on terms to incorporate the aforementioned effects. For example, Calamaz et al. [23] addeda strain softening term into the original JC plasticity model to describe the decrease in flow stressafter reaching a critical strain value. Later on, this modified plasticity model was further enhancedby Rotella and Umbrello [133] to phenomenologically merge the effect of microstructural changesinto material constitutive behavior. Apart from integrating the microstructure dependency, Cheng1286.2. Post-Machining DED Model Descriptionet al. [30] also incorporated the state of stress effect (stress triaxiality and lode dependency terms)into the original JC plasticity model. These modified versions of the original JC plasticity modelshave been successfully applied in metal cutting simulations. Nevertheless, no material plasticitymodels are available in the literature, which describe the materialâs flow stress by considering boththe stress states and microstructure features.Considering that both initial stress states and microstructure characteristics are critical tofeature the flow stress of DED-built materials, we propose a modified JC plasticity model (referredto as âthe MJC plasticity modelâ) that incorporates the cumulative effects of strain hardening,strain rate hardening, thermal softening, strain softening at large strains, and the state of stress onthe materialâs flow stress (ÏÂ¯MJC), as described in Eq. (6.1).ÏÂ¯MJC =[AJC +KHPâdg x+BJC ε¯nJCp(1exp(ε¯ap))]︸ ︷︷ ︸Term 1·[1 + CJC ln(ËÂ¯ÎµpËÂ¯Îµ0)]ï¸¸ ︷︷ ︸Term 2·[1â(T â T0Tm â T0)mJC]ï¸¸ ︷︷ ︸Term 3·ï£®ï£°D + (1âD)[tanh( 1ε¯p + P)R]Sï£¹ï£»ï¸¸ ︷︷ ︸Term 4· [1â cη(η â η0)]︸ ︷︷ ︸Term 5·ï£®ï£°csÎ¸ + (caxθ â csθ) · (fθ + 1fÎ¸)·ï£«ï£­Î³ â γ fθ+1fθfθ + 1ï£¶ï£¸ï£¹ï£»ï¸¸ ︷︷ ︸Term 6.(6.1)γ = 6.464[sec(Î¸¯Ï6)â 1]. (6.2)In Eq. (6.1), the MJC plasticity model includes six individual terms:(i) Term 1: The first term refers to the modified strain hardening term based on the originalstrain hardening term in the JC plasticity model. Specifically, in Term 1, instead of adopting aconstant initial yield strength (AJC), the Hall-Petch relationship [59, 131] is incorporated to updatethe change of the initial yield strength caused by the grain boundary strengthening effect. TheHall-Petch constant (KHP) is set as 298.4 MPa·µm1/2 [120], and the determination of dg x will bedetailed in Section 6.3.4. Additionally, the flow softening factor ( 1exp(ε¯ap)) is adopted from a modifiedJC plasticity model by Calamaz et al. [23]. The coefficient a is set as 2.0, which is taken from[145]. Other coefficients are the same as the original JC plasticity model, as listed in Table 6.1.(ii) Term 2 and 3: The second term (named the strain rate hardening term) and the thirdterm (known as the thermal softening term) are taken from the JC plasticity model.(iii) Term 4: The fourth term, also known as the TANH term, is used to describe thetemperature-dependent flow softening caused by dynamic recovery and dynamic recrystallization.This term was originally developed by Calamaz et al. [23] and later modified by Sima and O¨zel[145]. The coefficients in this term (D, P , R, S) are adopted from [128], which are set as 0.6, 0,1.0, and 5.0, respectively.1296.2. Post-Machining DED Model Description(iv) Term 5 and 6: These two terms are related to the states of stress effect, which are thefunctions of stress triaxiality (η) and Lode parameter (θ¯), respectively. These two terms were origi-nally developed by Bai and Wierzbicki [16], and further altered by Algarni et al. [5, 6]. Specifically,the fifth term is used to capture the effect of hydrostatic pressure on the flow stress. There are twocoefficients in this term: cη (featuring the hydrostatic pressure effects on material plasticity, whichis set as 0.40 [6]) and η0 (the reference stress triaxiality of the specimen in the calibration tests,which is defined as 0.333 [6]). Furthermore, the sixth term is selected to characterize the Lodeangle dependence on the materialâs plasticity. The parameter γ defined in Eq. (6.2) refers to thedifference in yield strength between von Mises and Tresca criteria in the deviatoric stress plane.The Lode dependence term is sensitive to three material constants (csθ, caxθ , and fθ). The values ofthese constants are taken from [6], which are set as 0.866, 1, and 0.75, respectively. Xu et al. [176]developed a material plasticity model that includes the above two terms (functions of η and θ¯)to capture the materialâs flow stress during orthogonal cutting of Ti6Al4V. Their findings suggestthat, for general cutting conditions, accounting for the in-process stress state (the variation in stressstates in the workpiece as the cutting tool advances) significantly improves the prediction accuracyfor cutting forces and chip geometries, compared to the original JC plasticity model. Nevertheless,all the workpieces in their cutting simulations were assumed to be free of the initial residual stress.Herein, the two terms (functions of η and θ¯) are combined to capture both the initial and in-processstress states on the flow stress of IN718 during post-machining processes. To achieve it, at first,a user-subroutine VUSDFLD is employed to read the stress components of each element at eachtime instant. Then, based on the calculation procedures listed in Appendix E, both η and θ¯ willbe computed at each integration point. As a result, with the tool advances, the flow stress at eachelement will be updated accordingly.To compare the difference in the materialâs flow stress between the two material plasticitymodels, the flow stress (ÏÂ¯) as a function of the equivalent plastic strain (ε¯p) are plotted at a materialpoint. This material point is located in IN718 structures, where the initial grain is assumed to beuniformly distributed, with an average value of dg x along the cutting velocity (x) direction. Forcomparison purposes, at this material point, only the initial residual stress component along thecutting velocity (x) direction (Ïre inxx ) is predefined to represent the stress state, with all other stresscomponents set to zero. Fig. 6.3 (a) and (b) compare the stress-strain (ÏÂ¯-Îµ¯p) curves at a materialpoint derived from both JC and MJC plasticity models at three initial stress statesâcompression(Ïre inxx = -500 MPa), shear (Ïre inxx = 0 MPa), and tension (Ïre inxx = 500 MPa)âunder both finergrain size (dg x = 10 µm) and coarser grain size (dg x = 100 µm) conditions, respectively. All thecurves are plotted at a temperature (T ) of 400 °C and a strain rate ( ËÂ¯Îµp) of 100 sâ1. As anticipated,the flow stress determined by the JC plasticity model shows no sensitivity to variations in eitherthe initial stress state or the built-in grain size values. In contrast, the flow stress obtained by theMJC plasticity model is highly sensitive to variations in both the initial stress states and grain sizevalues. Three features in the modified JC flow stress curves are selected to quantify the materialresponse under the given initial stress states and grain size values, including the initial flow stress1306.2. Post-Machining DED Model Description(the flow stress value when ε¯p is equal to zero), the peak flow stress (the maximum flow stressvalue), and the final flow stress (the flow stress value at a ε¯p of 10). As shown in Fig. 6.3 (c), thecompressive stress state features the highest flow stress value under both finer and coarser grainsize conditions, whereas the shear stress state presents the lowest flow stress value. Moreover, dueto grain size strengthening, finer grain conditions result in higher values in all three flow stressfeatures (including the initial flow stress, the peak flow stress, and the final flow stress) comparedto coarser grain conditions. Furthermore, considering the combined effects of initial stress statesand grain size on the materialâs flow stress, the initial flow stress values generated by the MJCplasticity model are higher than those obtained by the JC plasticity model (which stabilizes ataround 714.2 MPa).Figure 6.3: Comparison of the flow stress-strain curves between JC and MJC laws at a materialpoint under different initial stress states and having (a) dg x = 10 µm and (b) dg x = 100 µm. (c)shows a comparison of the key features in flow stress curves under the given initial stress statesand grain size values. Under the given initial tensile and free-of-stress states, the stress triaxialityterm (Term 5) yields the same value of 1. Therefore, the slight differences observed in these twostress-strain curves are attributed to the Lode dependency term (Term 6).1316.2. Post-Machining DED Model Description6.2.2.2 Material failure criteriaTo limit material deformation in metal cutting simulations, material plasticity models (herein, bothJC and MJC plasticity models) are usually combined with material failure criteria to characterizethe damage initiation [101] and evolution [157]. To identify the damage initiation, the most widelyused material failure criterion is the JC damage initiation model [75], where the material starts tolose its load-carrying capability when the equivalent plastic strain reaches ε¯p0,JC, as defined as inEq. (4.1). Table 6.2 lists the five JC damage parameters used in this study to describe the damageinitiation of DED-fabricated IN718 during post-machining simulations.Table 6.2: Coefficients of the JC damage initiation model for IN718 [44, 154, 174].d1 d2 d3 d4 d50.04 0.75 -1.45 0.04 0.89As a function of stress triaxiality (η), strain rate ( ËÂ¯Îµp), and temperature (T ), the JC damageinitiation model phenomenologically features the onset of material damage caused by nucleation,growth, and coalescence of voids. However, as recommended by Bai and Wierzbicki [16], bothstress triaxiality and Lode angle dependent terms need to be integrated into the damage initiationmodel to characterize the influence of stress states on the ductile fracture. To achieve it, underthe assumption of proportional loading and asymmetric metal plasticity, Bai and Wierzbicki [17]extended the stress-based Mohr-Coulomb (MC) fracture criterion [35] (previously used to predictfracture behavior for brittle materials) to describe failure behaviors for ductile materials. Themodified MC damage initiation model estimated the equivalent plastic strain at the onset of damageas a function of stress triaxiality (η) and Lode parameter (θ¯) to account for the stress state effects.The enhanced MC fracture model has been successfully used in various applications, particularlythe ductile fracture of metal sheets [5, 94]. Taking an investigation performed by Algarni et al.[5] as an example, in their study, monotonic tensile tests were performed on different types ofIN718 specimens with various loading conditions. By calibrating the enhanced MC fracture model,the fracture strains at damage initiation presented satisfactory agreements with the testing results.Nevertheless, all their experiments were conducted under quasi-static loading and room temperatureconditions. As a result, the effects of strain rate and thermal softening on ductile fracture werenot taken into account, which are critical for metal cutting simulations. To solve it, Sjo¨berg et al.[146] later calibrated the enhanced MC fracture model to feature the fracture behavior of IN718 athigh strain rates (varied from 1 to 1000 sâ1) and elevated temperature (ranging from 20 to 650 °C)conditions. Based on the above calibrations, da Silva et al. [34] employed the enhanced MC fracturemodel to predict the thermo-mechanical behavior of IN718 during orthogonal cutting simulations.The predicted cutting forces, temperature at the tool-chip interface, and chip morphologies agreedwell with the experiments.Herein, the extended MC fracture model by Bai and Wierzbicki [17] is further updated by1326.2. Post-Machining DED Model Descriptionadding the strain rate and thermal terms in the JC damage initiation model, to estimate thedamage behavior of DED-built IN718 samples. As a result, the modified MC damage initiationmodel (named the MMC damage initiation model, hereafter) calculates the equivalent plastic strainat the onset of damage (ε¯p0,MMC) by:ε¯p0,MMC ={c1[â1 + c223cos(Î¸¯Ï6)+ c2(η +13sin(Î¸¯Ï6))]}â1/nMMCÂ·[1 + d4 ln(ËÂ¯ÎµpËÂ¯Îµ0)]·[1 + d5(T â TrTm â Tr)],(6.3)where c1, c2, and nMMC are the coefficients in the modified MC damage initiation model by Baiand Wierzbicki [17], which are set as 1.936, 0.059, and 0.081, respectively, as listed in [5]; and thecoefficients d4 and d5 are the constants in the JC damage initiation model, as listed in Table 6.2.After damage initiation, a damage-induced stress softening model developed by Li andWierzbicki[94] (named the LW damage evolution model) is used to represent the damage evolution process.During the damage evolution process, the material progressively loses stiffness and dissipates thefracture energy. According to [94], the softened flow stress (ÏËflow) after damage initiation areexpressed as:ÏËflow = β · ÏÂ¯flow, (6.4)where ÏÂ¯flow denotes the originally undamaged flow stress without softening, which can be calculatedusing Eq. (2.25) for the JC plasticity model or Eq. (6.1) for the MJC plasticity model; and βrepresents the stress softening coefficient, which follows the power law, written as follows:β =(Dc âDtotalDc â 1)mLW, (6.5)where Dc and mLW are the softening parameters, which are set as 1.2 and 6 for IN718 [5], respec-tively; and Dtotal refers to the total damage variable, which averages all the damage mechanismsoccurring during fracture propagation and can be calculated by:Dtotal =â« ε¯p0dε¯pε¯p0, (6.6)where ε¯p0 is defined by Eq. (4.1) for the JC damage initiation model or Eq. (6.3) for the MMCdamage initiation model.In a classic FEM-based metal cutting model using the Lagrangian method, elements will bedeleted once the material stiffness is fully degraded. However, element deletion is not allowed inthe CEL-based metal cutting model, where the workpiece materials are supposed to flow withinthe fixed Eulerian elements. To avoid element deletion, the minimum β is limited to 0.4 [108]. Asa result, once Dtotal reaches 1.028, the material softening stops. Therefore, in these elements, thematerial will remain active and continue to bear thermo-mechanical loads with a residual stiffness1336.2. Post-Machining DED Model Descriptionthroughout the simulation. Consequently, combining the material plasticity, damage initiation,and evolution models, the equivalent flow stress (ÏÂ¯) of IN718 in the post-machining simulationswill follow:ÏÂ¯ =ï£±ï£´ï£´ï£´ï£²ï£´ï£´ï£´ï£³ÏÂ¯flow 0 â¤ Dtotal â¤ 1,ÏËflow = β · ÏÂ¯flow 1 < Dtotal < 1.028,¯ËÏflow = 0.4 · ÏÂ¯flow Dtotal â¥ 1.028.(6.7)Based on the above analyses, two combinations of material constitutive laws will be assessed,which are listed as follows:(1) Law 1: The JC plasticity model, with the JC damage initiation model and the LW damageevolution model;(2) Law 2: The MJC plasticity model, with the MMC damage initiation model and the LWdamage evolution model.As shown in Fig. 6.4, notable differences can be observed between the two material constitutivelaws in capturing the plastic and damage behaviors of IN718 at various initial stress states (repre-sented by η and θ¯). Analogous to the results presented in Fig. 6.3, all the calculations in Fig. 6.4were performed at an initial grain size (dg x) of 10 µm, a temperature (T ) of 400 °C, and a strainrate ( ËÂ¯Îµp) of 100 sâ1. Under the given η and θ¯ (both varying from -1 to 1), Fig. 6.4 (a) and (b)show the equivalent plastic strain at damage initiation (ε¯p0) calculated by Law 1 and Law 2, whichare labeled as Surface a and Surface b, respectively. It can be seen that no matter how θ¯ is, ε¯p0reduces as η increases. This observation aligns with the tensile test results reported in [16, 22, 78].However, when η is fixed, the changes in ε¯p0 with respect to θ¯ vary depending on the selected ma-terial constitutive law. Specifically, as θ¯ increases from -1 to 1, ε¯p0 can be calculated by Law 1 isconsistently constant, whereas ε¯p0 determined by Law 2 exhibits a decreasing-then-increasing trend.To quantify how ε¯p0 varies with θ¯, three distinct planes at η = -0.5 (Plane c), η = 0 (Plane d),and η = 1 (Plane e) were selected. The intersection lines between the damage initiation surfaces(Surface a and Surface b) and the three planes (Plane c, Plane d, and Plane e) correspond to theθ¯-ε¯p0 curves at η = -0.5, η = 0, and η = 1, respectively, as depicted in Fig. 6.4 (c), (d), and (e).It can be seen that for each η, the intersection points of the two θ¯-ε¯p0 curves, calculated by Law 1and Law 2, share the same ε¯p0 at θ¯ = -0.75 and 0.88. This indicates that, no matter θ¯, when thestress states of IN718 are characterized by θ¯ = -0.75 or 0.88, the two material constitutive lawsagree on the starting point of damage-induced stress softening. Nevertheless, when θ¯ varies from-0.75 to 0.88, ε¯p0 estimated by Law 1 is higher than that calculated by Law 2, with the maximumdeviation occurring at θ¯ = 0. Meanwhile, for stress states where θ¯ is greater than 0.88 or lessthan -0.75, the material is more prone to damage using Law 1 compared to using Law 2. Moreimportantly, as η increases from -1 to 1, the above deviations in the predicted ε¯p0 between the twomaterial constitutive laws decrease. Due to the above discrepancy in the materialâs flow stress, thefinal residual stress may vary depending on the selected material constitutive laws.1346.2. Post-Machining DED Model DescriptionFigure 6.4: The equivalent plastic strains at the onset of damage are predicted by Law 1 (a) andLaw 2 (b). For a detailed illustration, the θ¯-ε¯p0 curves are plotted under (c) η = -0.5, (d) η = 0,and (e) η = 1, respectively, together with the corresponding Îµ¯p-ÏÂ¯ profiles for three cases (θ¯= -1, 0,and 0.88) at each η.1356.3. Experiments and Validation6.3 Experiments and Validation6.3.1 DED experimentsSpherical IN718 powders (Praxair Surface Technologies, USA) with an average diameter of 72.1µm were selected as the feeding material. A commercial DED machine (Optomec LENS CS-250,USA) was used to fabricate the single-track IN718 samples. This machine was equipped with a 1kW nLight laser system, and a 2 mm laser spot size was selected. To enhance the metallurgicalbonding between the powder and the substrate, IN718 plates were chosen as the substrates. Theseplates were annealed, having a yield strength of 482.6 MPa and a Rockwell hardness of B95. Thedimensions of IN718 substrates are 80 mm (length) Ã 6 mm (height) Ã 6 mm (width). Beforedeposition, these substrates were treated by a sandblaster and then cleaned and degreased withethanol. In the fabrication of single-track IN718 samples, the deposition parameters are listed asfollows: a laser power of 800 W, a laser traverse speed of 508 mm/min, and a powder feed rate of10 g/min. Argon gas served as the carrier and protective gas, with a shielding gas flow of 9 L/min.6.3.2 Post-machining experimentsOrthogonal cutting tests of DED-built single-track IN718 samples were carried out using a three-axis CNC milling machine (Fadal VMC 2216 A). This machine has a maximum rapid feed rate ofabout 17.8 m/min along the x direction. The cutting tools were uncoated carbide inserts (SandvikCoromant TPUN 221 H13A). A laser confocal microscope was used to measure the toolâs micro-geometry. Based on the measurements, the tool edge radius (re) was 15 ± 1 µm, while the nominalrake and clearance angles were 0 and 11 degrees, respectively. As illustrated in Fig. 6.5, the insertwas mounted on a tool holder (Sandvik Coromant 177.9-2009-11). The assembled tool holder wasthen attached to the machineâs spindle using a tool holder adapter. The angular movements ofthe spindle were locked throughout the cutting experiments. Since the tool holder has a 0-degreelead angle and a 6-degree rake angle, after mounting the insert into the tool holder, the resultanttool angles were 6 degrees for the rake angle (γo) and 5 degrees for the clearance angle (αn)during cutting experiments, which will be used in the post-machining DED simulation. Each IN718single-track sample underwent post-machining through a two-step orthogonal cutting experiment,including roughing and finishing, under dry cutting conditions. The roughing operations aimed toensure the consistency of the uncut chip thickness in the finishing process, while producing a post-machined surface with a sufficient width (greater than 1 mm) for the subsequent residual stressmeasurements. This necessity arises due to the âsemi-ellipticalâ shape of the single-track sampleâscladding profiles in the transverse cross-section, parallel to the x-z plane. We have examinedthe surfaces of IN718 single-track samples before and after post-machining using a confocal laserscanning microscope (Olympus LEXT OLS3100) and a scanning electron microscope (SEM, HitachiSU3500). As presented in Fig. 6.6(a), before roughing, the average width of the IN718 sample variesfrom zero (near the top area of the deposit) to around 2.90 mm (at the substrate-deposit interface).The roughing processes end until the width of the roughing surface reaches around 1.45 mm, after1366.3. Experiments and Validationwhich the finishing operation starts. Table 6.3 summarizes the cutting parameters used in bothroughing and finishing processes, where the width of cut in the finishing process referred to theaverage width between the roughing surface (presented in Fig. 6.6(c)) and the finishing surface(shown in Fig. 6.6(e)). During finishing, both tangential forces (Ft) and feed forces (Ff ) weremeasured using a force dynamometer (Kistler 9257B).Figure 6.5: Experimental setup in orthogonal cutting of single-track IN718 samples.Table 6.3: Orthogonal cutting parameters.Cutting velocity Vc Uncut chip thickness H Width of cut wRoughing 15 m/min 0.05 mm -Finishing 15 m/min 0.12 mm 1.6 ± 0.2 mm1376.3. Experiments and ValidationFigure 6.6: Laser confocal and SEM images of the single-track IN718 samples before and after thepost-machining at different locations: (a) As-deposited surface; (b) Transition region between theas-deposited surface and the roughing surface; (c) Machined surface after roughing; (d) Transitionregion between the roughing surface and the finishing surface; (e) Machined surface after finishing.1386.3. Experiments and Validation6.3.3 Residual stress measurements and reconstructionResidual stresses along the longitudinal (x) and the transverse (y) directions were measured byan X-ray diffractometer based on the sine-squared-psi (Ψ) method. Details related to the residualstress measurement conditions are listed in Table 6.4. In addition, to determine the residual stressdistribution beneath the machined surface, an electrolyte jet machining process [3] was applied toincrementally remove a specific amount of material under both roughing and finished surfaces.Table 6.4: Details of residual stress measurement conditions for DED-built single-track IN718samples after orthogonal cutting.Items DescriptionX-ray source Mn K alpha (5.9 KeV)X-ray tube power 20 kV and 4 mACollimator shape and size Round, Ï1 mmDiffraction plane {311}Brag angle 155°Number of tilts 11 (from -30° to 30°)Exposure time 10 secondsX-ray elastic constants (1/2)S2: 6.33Ã 10â6 MPaâ1,âS1: 1.42Ã 10â6 MPaâ1Considering the incompleteness of residual stress data that includes only two stress compo-nents at each measurement point provided by XRD, an iterative stress reconstruction algorithm,as developed by Sunny et al. [154], is used to estimate the remaining stress components at eachmeasurement point and all stress components between measurements points, to ensure the com-pleteness and compatibility of the initial residual stress fields in the post-machining model. This isnecessary because incomplete residual stress tensors at each measurement point (comprising of onlytwo stress components) are insufficient for determining the stress state parameters (stress triaxialityη and Lode parameter θ¯), which are required for describing the materialâs flow stress in the orthog-onal cutting simulation, especially when the initial stress states are incorporated. Alternatively,linear interpolation could be used to calculate residual stress values between measurement points;however, this approach may produce interpolated stresses that are not fully compatible with themeasured values.1396.3. Experiments and ValidationFigure 6.7: Residual stress reconstruction principles and results: (a) A schematic diagram toillustrate the principle of the stress reconstruction algorithm; (b) Reconstructed residual stressdistribution along the x direction; (c) Comparison of measured residual stress in the x directionand the reconstructed residual stress profile; (d) Comparison of measured residual stress in the ydirection and the reconstructed residual stress profile.1406.3. Experiments and ValidationFig. 6.7 (a) presents the principle of the stress reconstruction algorithm. Initially, neutral stressstates are predefined throughout the whole structure, except at the focal elements (Em), where theirresidual stress values are determined by XRD. The locations of these elements are identified, andtheir corresponding stress values (ÏEmij , referred to as the âknownâ stress states) are assigned basedon XRD measurement values. This assignment creates an incompatible stress state for the wholestructure. Then, the first static equilibration will be executed to update the stress states withinthe structure by simultaneously solving compatibility and equilibrium equations under the givenboundary conditions [154]. As a result, a compatible residual stress field over the entire structurecan be computed via multiple static equilibrium iterations: taking the resulting stress distributionfor the entire structure from the previous iteration, re-applying the âknownâ stress states in thefocal elements, and finding the new equilibrium states. The above iterative process ends once theabsolute difference between the computed values at the focal elements and the measured valuesat the corresponding locations is less than the uncertainties in XRD measurements (presented asmeasurement error bars in Fig. 6.7 (c) and (d)). Once convergence is achieved, the final residualstress distribution represents the reconstructed stress field for the given structure. Following theabove procedures, we employed the measured residual stress values in both x- and y-directions toreconstruct the initial stress states within the DED-built IN718 sample after roughing. Fig. 6.7(b) illustrates the reconstructed residual stress distribution in the x-direction, which requires atotal number of static equilibrium iterations of 7. Meanwhile, as shown in Fig. 6.7 (c) and (d),the normalized root mean squared errors are 27.7% and 30.0% between measured residual stressvalues and the corresponding residual stress profiles after reconstruction in the x- and y-direction,respectively. These reconstructed residual stress data will be fed into the post-machining DEDmodel shown in Fig. 6.2 and assigned as its initial stress conditions through a user-subroutineSIGINI [63].6.3.4 Grain size measurements and analysisThe grain size distribution in DED-built single-track IN718 samples was characterized by electronbackscatter diffraction (EBSD), using a focused ion beam SEM (Helios NanoLab 650). Before it,a multiple-step sample preparation procedure was required. At first, single-track IN718 sampleswere sectioned by WEDM. Then, a 2-component copper-based cold embedding agent was usedto mount the WEDM-produced pieces in plastic molds under a 3:2 mixture of Technovit® 5000powder (KULZER) and Technovit® 5000 liquid (KULZER). The mounted samples were groundby silicon carbide waterproof metallurgical papers (Allied High Tech), with 400, 600, 800, and 1200grits in turn. After that, using water-based monocrystalline diamond suspensions (Allied HighTech), the ground surfaces were sequentially polished by 6 µm, 3 µm, 1 µm, and 0.5 µm diamondfilms (Struers). Finally, to remove the micro-defects, the polished surfaces were further preparedby non-crystallizing colloidal silica polishing suspensions under a vibratory polisher (Buehler) for2-3 hours.To determine dg x and dg y, the DED-generated microstructure needs to be reconstructed. As1416.3. Experiments and Validationmentioned, dg x represents the average grain size along the cutting velocity (x) direction, which isused to update the initial yield strength of DED-created IN718 samples in Eq. (6.1). Meanwhile,dg y is equal to the average grain size along the width of cut (y) direction, as a reference to definethe thickness of the FEM model in the y-direction, as mentioned in Section 6.2.1. To characterizethe actual grain size distribution within the DED-created components, we conducted EBSD testson three representative planes: the xoy, xoz, and yoz planes, referred to as the âTop viewâ, âSideviewâ, and âFront viewâ, respectively. Before the EBSD measurement, the above three planes weresectioned by WEDM and underwent metallurgical treatments, as detailed in Section 6.3.3. Duringthe measurement, the accelerating voltage was set as 30 kV and a 500Ã500 µm2 area was scannedat a step size of 4 µm in each plane.The obtained EBSD data were analyzed by EDAX OIM Analysisâ¢ software to characterizethe grain size information. Before that, a âclean-upâ procedure of the EBSD data was performedto remove isolated pixels (wild spike removal) and pixels with invalid orientation (zero solutionremoval). The selected phase for plotting the inverse pole figure (IPF) was the chromium-nickelphase. The misorientation angles ranging from 2° to 15° and greater than 15° were defined as thelow-angle grain boundary (LAGB) and the high-angle grain boundary (HAGB), respectively. Theidentified grain boundaries were then fitted as ellipses to obtain elliptical-based shape parameters(the mean major diameter of Lmajor, the mean minor diameter of Lminor, and the mean rotationangle of Îrotation) in each plane. Following the above post-processing procedures, Fig. 6.8 (a), (b),and (c) present the grain boundary maps, IPF, and the distribution of elliptical-fitted grain shapeparameters (major diameters, minor diameters, and rotation angles) in xoy, xoz, and yoz planes,respectively. The results suggest that columnar grains are dominant within the DED-fabricatedsingle-track IN718 sample.To demonstrate how to calculate dg x and dg y, as illustrated in Fig. 6.8 (d), the simplifiedgrains are sectioned in xoy, xoz, and yoz planes, with the profiles in each plane all shaped asellipses. Therefore, based on the average grain shape parameters fitted by ellipses in the xoy planeor the xoz plane, dg x can be determined by calculating the intercept of the corresponding ellipseswith the x-axis, written as:dg x =ââââ L2major · L2minorL2minor · cos2Îrotation + L2major · sin2Îrotation, (6.8)Analogue to dg x, dg y refers to the intercepts of the ellipses in the xoy plane or the yoz planewith the y-axis, which can be expressed as:dg y =ââââ L2major · L2minorL2major · cos2Îrotation + L2minor · sin2Îrotation, (6.9)1426.3. Experiments and ValidationFigure 6.8: Grain size information of DED-built single-track IN718 sample characterized byEBSD. (a), (b), and (c) refer to the grain boundary maps, IPF, and the distribution of fitted grainshape parameters (major diameters, minor diameters, and rotation angles) in xoy, xoz, and yozplanes, respectively. (d) illustrates how to calculate dg x and/or dg y based on the average ellipsegrain shape parameters identified in each plane.1436.3. Experiments and ValidationBased on the above calculation procedures, three EBSD tests were conducted in each planementioned above (xoy, xoz, and yoz planes). The resulting dg x and dg y values at each plane andtheir mean values are listed in Table 6.5. Notably, the values of dg x and dg y derived from Fig. 6.8(a), (b), and (c) are termed as Top view No. 1, Side view No. 1, and Front view No. 1, respectively.The mean values of dg x and dg y will be used in the post-machining DED model.Table 6.5: dg x and dg y values in each plane and their corresponding mean values.No. dg x [µm] dg y [µm]Top view 1 50.7 84.22 49.6 87.93 58.7 79.0Side view 1 39.1 -2 45.8 -3 43.3 -Front view 1 - 127.92 - 92.13 - 131.3Mean values 47.9 100.46.3.5 Model validation and discussionsThis section assesses the effectiveness of the two material laws (Law 1 and Law 2, introduced inSection 6.2.2) in capturing the thermo-mechanical response of DED-built IN718 materials duringorthogonal cutting processes. This assessment is performed by comparing the simulated cuttingforces and finished surface residual stress profiles with experimental measurements.Table 6.6 lists the measured tangential (Ft) and feed (Ff ) cutting forces per unit width withthose simulated using the two material laws (Law 1 and Law 2). It is suggested that the predictedFt aligns well with the measurements, with the maximum mean square error of less than 14%(obtained under Law 1). However, the predicted Ff is underestimated, showing mean square errorsof 45.4% for Law 1 and 27.9% for Law 2. The underestimation of feed force can be attributed tothe neglection of the initial tool wear effect. In metal cutting experiments, rapid changes in tooledge geometries occur during the initial cutting periods, which increases the feed forces, especiallywhen a fresh insert is used. As reported by Laakso et al.[88], the initial flank wear contributes20-50% of the feed force, even without considering friction. This work focuses on determining thecumulative effects of initial residual stress states and microstructure features on the final finishedsurface residual stress after post-machining; therefore, the initial tool wear effect has not beenintegrated into the post-machining simulations. Apart from the neglection of initial tool wear1446.3. Experiments and Validationeffects, as demonstrated in [183], the unmatched feed forces may be linked to the simplified frictioncoefficient used at the tool-workpiece interface in the post-machining DED model. It should alsobe noted that under the identical post-machining parameters, the cutting forces predicted by Law2 are higher than those predicted by Law 1. This can be attributed to the differences in the initialyield strength employed by the two material laws. As discussed in Section 6.2.2, compared toLaw 1, which adopts a constant initial yield strength of AJC, Law 2 incorporates the Hall-Petchrelationship to estimate the increase in the initial yield strength caused by the grain boundarystrengthening effect, leading to higher predicted cutting forces.Table 6.6: Comparison between measured and simulated cutting forces under Vc = 15 m/min andH = 0.12 mm.Measurement Simulation - Law 1(error)Simulation - Law 2(error)Ft [N/mm] 404±41 348±4 422±23(-13.9%) (+4.5%)Ff [N/mm] 262±33 143±4 189±10(-45.4%) (-27.9%)Fig. 6.9 (a) and (b) show the predicted residual stress distribution along the cutting velocity(x) direction (Ïre finalxx ) by Law 1 and Law 2, respectively. For comparison, as shown in Fig.6.9 (a), we selected 15 paths to output the final residual stress profiles on finished surfaces. Aspresented in Fig. 6.9 (c) and (d), the simulated residual stress profiles along the x and y directions(Ïre finalxx and Ïre finalyy ) are compared with the measured values, respectively. It can be observedthat under the given cutting conditions, Law 1 and Law 2 are both effective in estimating thepost-machining penetration depth, which refers to the location where the post-machining-inducedresidual stress begins to align with the initial stress profiles. Specifically, the XRD measurementresults indicate that the final residual stress starts to follow the initial profiles at a depth of about230 µm below the finished surface. Meanwhile, predictions from the two material laws suggest apost-machining penetration depth of around 194 µm beneath the finished surface. Post-machiningpenetration depth defines the boundary beyond which the thermo-mechanical loads exerted by thecutting tool on the finished surface become negligible. This slightly shallower penetration depth inthe predictions, compared to the experiments, can be explained by the underestimation of cuttingforces (specifically the feed forces) in the post-machining simulations.1456.3. Experiments and ValidationFigure 6.9: Predicted residual stress distribution in the cutting velocity (x) direction under: (a)Law 1; (b) Law 2. Comparison between experimentally measured and numerically predicted residualstress profiles: (c) Residual stresses in the x direction; (d) Residual stresses in the y direction.Furthermore, although the predicted residual stresses present tensile states on finished surfaces,their magnitudes and peak locations vary depending on the material laws. Specifically, Law 2exhibits lower tensile residual stress values on the finished surface compared to Law 1 in both x-and y-directions. Moreover, under Law 1, the maximum tensile residual stress occurs at the finishedsurface. However, when Law 2 is employed, the predicted tensile residual stress peaks at the locationbelow the finished surface, which aligns with the measurements. As discussed in [108] and [183], inthe dry machining simulation of nickel-based alloys, using the JC constitutive law (referred to asLaw 1) always results in the maximum tensile residual occurring on the machined surface. This is1466.4. Quantitative Analyses of Surface Residual Stress after Post-Machining DEDbecause Law 1 only accounts for the generation of tensile residual stress by two factors: the tooltraction on the newly generated surface (mechanical effect) and the restricted thermal shrinkageof the surface material by the surrounding material (thermal effects). Compared to the subsurfacematerial, due to the direct contact with the cutting tool, surface material experiences more severeplastic deformation and higher temperature during the cutting processes. Consequently, aftercooling down, the predicted residual stress by Law 1 consistently peaks on the finished surfaces inboth x and y directions. However, as detailed in Section 6.2.2, Law 2 introduces a TANH term todescribe the change of materialsâ flow stress due to dynamic recovery and dynamic recrystallization.Liu et al. [100] reported that dynamic recovery is unavoidable during the thermal processing ofnickel-based alloys, which is accompanied by partial annihilation of dislocations and a reduction inresidual stress. Moreover, Smith et al. [147] summarized that in highly strained microstructures,dynamic recrystallization occurs to relieve residual stress by promoting the formation of strain-free grains. As discussed in [183], the surface material experiences the highest plastic strain andtemperature, which gradually decreases as the depth from the finished surface increases. As aresult, the tensile residual stress, relaxed by dynamic recovery and dynamic recrystallization, alsodiminishes progressively as the depth increases. This residual stress relaxation shapes the finalresidual stress profiles predicted by Law 2. Therefore, the finished surface residual stress becomesless tensile and the location of the peak tensile residual stress shifts to the subsurface.Overall, the predicted cutting forces and finished surface residual stress profiles show agreementwith measurements, especially for Law 2. To quantitatively determine the cumulative interactionsof initial residual stress states and microstructure features on the final residual stress on finishedsurfaces, in the next section, the developed post-machining DED model is employed to simulatethe thermo-mechanical response of IN718 under hypothetical stress states and grain size scenarios.6.4 Quantitative Analyses of Surface Residual Stress afterPost-Machining DEDThis section focuses on quantifying how initial residual stress and grain size values contribute tofinal residual stress on post-machined surfaces of DED components. The aim is to understand theformation mechanism of surface residual stress on post-machined DED parts, providing insightsinto post-machining and/or other post-treatment techniques for achieving enhanced performanceof DED-fabricated metallic components.To evaluate the post-machining DED processes comprehensively, this section examines thethermo-mechanical response of IN718 under a series of 9 orthogonal cutting simulations. Thesesimulations incorporate two material laws (Law 1 and Law 2, discussed in Section 6.2.2), threeinitial residual stress states (compressive-dominant, zero-stress, and tensile-dominant) in the chipformation region, and two initial grain size values (dg x = 10 µm and 100 µm). Fig. 6.10 de-tails the designed initial residual stress distribution under the given workpiece geometry and theircorresponding stress reconstruction results (based on the algorithm discussed in Section 6.3.3).1476.4. Quantitative Analyses of Surface Residual Stress after Post-Machining DEDSpecifically, at the locations ahead of the cutting tool, the designed residual stress along the cut-ting velocity (x) direction (Ïre inxx ) was set to -500 MPa for workpieces dominated by compressiveresidual stress (Initial profile 1), 0 MPa for free-of-stress workpieces (Initial profile 2), and 500 MPafor workpieces dominated by tensile residual stress (Initial profile 3). The reconstructed residualstress fields were set as the initial stress conditions in the post-machining DED model. As listedin Table 6.7, all the post-machining simulations of IN718 were conducted under a cutting velocity(Vc) of 45 m/min and an uncut chip thickness (H) of 0.1 mm. After completing all the simulations,the resulting cutting forces per unit width and final surface residual stress profiles were compared,and their differences were analyzed.Figure 6.10: (a-c) Schematic diagrams illustrating the designed initial residual stress distributionwithin the workpiece featuring (a) compressive-dominant, (b) zero, and (c) tensile-dominant stressstates in the chip formation region. The corresponding designed and reconstructed residual stressprofiles at the selected path are compared, which are referred to as (d) Initial profile 1, (e) Initialprofile 2, and (f) Initial profile 3, respectively. The two expanded areas are used to balance thedesigned initial residual stress states. As a result, after completing the residual stress reconstruction,the residual stress distribution in the middle area of the workpiece will remain at the designed values,while the residual stress states in the whole structure achieve both equilibrium and compatibility.1486.4. Quantitative Analyses of Surface Residual Stress after Post-Machining DEDTable 6.7: Details about 9 post-machining simulations of IN718.Case No. Constitutive laws Stress profiles dg x [µm]1 Law 1 Initial profile 1 -2 Law 1 Initial profile 2 -3 Law 1 Initial profile 3 -4 Law 2 Initial profile 1 105 Law 2 Initial profile 2 106 Law 2 Initial profile 3 107 Law 2 Initial profile 1 1008 Law 2 Initial profile 2 1009 Law 2 Initial profile 3 1006.4.1 Cutting forcesUnder the 9 post-machining DED simulation conditions shown in Fig. 6.11(a), Fig. 6.11(b) presentsthe simulated tangential (Ft) and feed (Ff ) cutting forces per unit width. Under Law 1 (Cases1-3), the simulated Ft and Ff are at around 258.4 N/mm and 83.2 N/mm, respectively. When Law2 is applied with a finer initial grain size (dg x) of 10 µm (Cases 4-6), the simulated Ft and Ff areabout 305.2 N/mm and 102.0 N/mm, respectively. As dg x increases to 100 µm (Cases 7-9), Ftand Ff decrease slightly to approximately 290.8 N/mm and 90.3 N/mm, respectively. The aboveresults highlight two key findings. First, regardless of whether material laws (Law 1 or Law 2) areemployed, variations in the initial residual stress states have no effects on cutting forces. Second,the initial grain size values, particularly under Law 2, play a critical role in determining cuttingforces. Since [70] details the mechanism by which a reduction in grain size leads to an increase inthe average cutting forces, the following discussion will focus on revealing the first key finding.As a key indicator of material elastoplastic behavior around the cutting tool, cutting forcesreflect the required energy for material deformation and removal in metal cutting processes. Toexplain why cutting forces remain unaffected by the initial residual stress states, we analyzed thestress fields along the cutting velocity (x) direction (Ïxx) in the workpiece material ahead of thecutting tool. Fig. 6.12 (a)-(i) presents the distributions of Ïxx during the orthogonal cutting processof IN718 at t = 1.35 ms for each of the 9 cases (Cases 1-9). Despite the initial residual stress states(Ïre inxx ) along the cutting velocity (x) direction ahead of the cutting tool varying from compressiveto tensile across Cases 1 to 3, Cases 4 to 6, and Cases 7 to 9, stresses (Ïxx) in the workpiecematerial at three shear zones consistently show a compressive-dominant state in all cases. Apartfrom the compressive stress caused by phase transformation, as discussed in [183], the presence ofcompressive stress ahead of the cutting tool arises from two primary factors: material compression1496.4. Quantitative Analyses of Surface Residual Stress after Post-Machining DEDby the moving cutting tool (mechanical effect) and constrained thermal expansion of the workpiecematerial around the cutting tool caused by the restriction of surrounding materials (thermal effect).To fully understand the material response under different initial residual stress states, a point(referred to as âFocal point 1â) within the chip formation area in the steady-state cutting period isselected to record the temperature (TEMP), the equivalent plastic strain (PEEQ), the stress alongthe cutting velocity (x) direction (Ïxx), and the stress state parameters (η and θ¯) during the 9post-machining simulations. As shown in Fig. 6.13 (a), (d), and (g), the evolution of TEMP andPEEQ follows the same profiles across Cases 1 to 3, Cases 4 to 6, and Cases 7 to 9, respectively.These profiles exhibit rapid increases from the initial valuesâroom temperature for TEMP andzero for PEEQâuntil reaching the same peak values when the focal point contacts the cuttingtool (at around 1.35 ms). In addition, the profiles of η and θ¯ shown in Fig. 6.13 (c), (f), and (i)also reveal the convergence of η and θ¯ at the focal point during post-machining across Cases 1 to3, Cases 4 to 6, and Cases 7 to 9, respectively. Under Cases 4-6, the distribution of η and θ¯ inthe workpiece at different post-machining time increments (t = 0 ms, 0.3 ms, 0.825 ms, and 1.35ms) are presented in Fig. 6.14 (a)-(c) and Fig. 6.14 (d)-(f), respectively. The results suggest thatregardless of the initial stress states (the initial values of η and θ¯), as the post-machining processprogresses, η at the focal point gradually transitions into a negative value. Conversely, θ¯ initiallyshifts to a negative value but finally converges to a positive value. As demonstrated in [16], ηplays a dominant role in determining the stress states within the workpiece. As long as the η isnegative/positive, no matter how θ¯ is, the material will be in the compressive/tensile state. Asa result, although the initial values of Ïxx in the chip formation region vary from compressive totensile across Cases 1 to 3, Cases 4 to 6, and Cases 7 to 9, they all converge to the same negativevalue (compressive stress states), as presented in Fig. 6.13 (b), (e), and (h), respectively.The above analyses suggest that a uniform thermo-mechanical response of workpiece materialoccurs in the chip formation area, no matter how the initial state of stress is. This uniformityin stress distribution in front of the cutting tool indicates that tool advancement during post-machining processes generates a localized compressive-dominant stress concentration region, whichpredominates over and effectively âresetsâ any pre-existing residual stress fields around the cuttingtool. Consequently, there are negligible differences in cutting forces across different initial stressconditions.1506.4. Quantitative Analyses of Surface Residual Stress after Post-Machining DEDFigure 6.11: (a) Schematic representation of initial residual stress distributions, grain size values,and material laws across 9 orthogonal cutting simulation cases for IN718. (b) Simulated tangential(Ft) and feed (Ff ) forces per unit width in each case. The color scale in (a) provides a quantitativerepresentation of the initial residual stress distribution in MPa within the workpiece and is usedconsistently throughout the subsequent figures.1516.4. Quantitative Analyses of Surface Residual Stress after Post-Machining DEDFigure 6.12: Workpiece stress distribution during orthogonal cutting of IN718 at t = 1.35 ms for(a) Case 1; (b) Case 2; (c) Case 3; (d) Case 4; (e) Case 5; (f) Case 6; (g) Case 7; (h) Case 8; (i)Case 9.1526.4. Quantitative Analyses of Surface Residual Stress after Post-Machining DEDFigure 6.13: Evolution of temperature (TEMP), equivalent plastic strain (PEEQ), stress alongthe cutting velocity (x) direction (Ïxx), and state of stress parameters (η and θ¯) at a selectedpoint within the chip formation region during 9 post-machining simulations: Subplots (a), (d) and(e) represent the TEMP and PEEQ curves for Cases 1-3, Cases 4-6, and Cases 7-9, respectively;Subplots (b), (e), and (h) depict the Ïxx profiles for Cases 1-3, Cases 4-6, and Cases 7-9, respectively;Subplots (c), (f), and (i) show η and θ¯ traces for Cases 1-3, Cases 4-6, and Cases 7-9, respectively.1536.4. Quantitative Analyses of Surface Residual Stress after Post-Machining DEDFigure 6.14: Distribution of η and θ¯ within the workpiece during post-machining simulationsunder the given time instants: Subfigures (a)-(c) represent the variation of η in the workpiece atCases 4-6, respectively; Subfigures (d)-(f) depict the variation of θ¯ in the workpiece at Cases 4-6,respectively.6.4.2 Final surface residual stressFig. 6.15 presents three scenarios of initial residual stress (Ïre inxx ) distribution along the cuttingvelocity (x) direction in the âoriginal workpieceâ, and their corresponding final residual stress(Ïre finalxx ) maps in the âfinal workpieceâ after completing the 9 post-machining simulation cases. Itcan be observed that under Law 1, the final surfaces are always featured by tensile residual stress1546.4. Quantitative Analyses of Surface Residual Stress after Post-Machining DEDalong the cutting velocity (x) direction. In contrast, when Law 2 is applied, Ïre finalxx on all finalsurfaces remains compressive.As illustrated in Fig. 6.15 (a), 15 paths were selected to extract the residual stress profiles alongthe cutting velocity (x) direction before and after post-machining simulations, and the averageprofiles are presented in Fig. 6.16. As demonstrated in Fig. 6.16, regardless of whether theinitial residual stress (Ïre inxx ) in the material removal region is dominated by compressive (Case1), zero-stress (Case 2), or tensile (Case 3) states, applying Law 1 always produces tensile residualstress values (Ïre finalxx ) of around 460 MPa on finished surfaces. However, Law 2 reflects variationsin compressive residual stress on the finished surfaces, with magnitudes depending on the initialresidual stress states and grain size values. Specifically, when dg x is 10 µm, in the post-machiningof regions dominated by compressive residual stress (Case 4), the finished surface residual stress(Ïre finalxx ) is about -723 MPa. Under the same dg x, as the initial residual stress states (Ïre inxx ) shiftfrom compressive (Case 4) to zero-stress (Case 5) or tensile (Case 6), Ïre finalxx on finished surfacesreduce to around -658 MPa (Case 5) and -572 MPa (Case 6), respectively. Furthermore, as dg xincreases to 100 µm, after post-machining of regions initially dominated by compressive (Case 7),zero-stress (Case 8), and tensile (Case 9) residual stress states, Ïre finalxx on post-machined surfacesdecreases to approximately -551 MPa, -375 MPa, and -263 MPa, respectively.To understand the formation of residual stress on post-machined surfaces, we chose a point onfinished surfaces (named âFocal point 2â, as illustrated in Fig. 6.13) in the steady-state cuttingregion to output the temperature (TEMP), stress along the cutting velocity (x) direction (Ïxx), thestate of stress parameters (η and θ¯), and plastic strain along the x direction (PExx). In addition,to quantify the thermo-mechanical loads acted by the cutting tool during post-machining, the localshear and normal contact forces on the cutting tool below the stagnation point were calculated,and their evolution curves were presented in Fig. 6.18. The stagnation point is identified as thelocation on the cutting tool where the normal contact forces reach their peak values while the shearcontact forces are zero, as described in [182] and [183].As shown in Fig. 6.17 (a), regardless of whether the initial stress states on the finished surfacesare compressive-dominant (Case 1), free-of-stress (Case 2), or tensile-dominant (Case 3), as thecutting tool advances, Ïxx on finished surfaces gradually converges, reaching peak compressivevalues once the cutting tool approaches âFocal point 2â. This is aligned with stress evolution curvesin the chip formation region, as discussed in Section 6.4.1. Once the cutting tool moves away, Ïxxrapidly transitions from compressive to tensile. Except for phase transformation, the formation oftensile stress on the finished surface results from the traction by the tool flank face on the newlyfinished surface (mechanical effects) and the restricted thermal shrinkage of the finished surfacematerial constrained by the surrounding bulk material during cooling down (thermal effects). Fig.6.18 (b), (c), and (d) depicts the evolution of local contact forces on the cutting tool below thestagnation points for Cases 1, 2, and 3, respectively. The results suggest that all three cases (Cases1 to 3) share the same average values for the local normal and shear contact forces (the sum ofthe contact forces on the cutting tool below the stagnation point) of 21.7 N/mm and 3.0 N/mm,1556.4. Quantitative Analyses of Surface Residual Stress after Post-Machining DEDrespectively. This demonstrates that, no matter how the initial residual stresses are distributedwithin the workpiece, the thermo-mechanical loads exerted by the cutting tool on the finishedsurfaces are identical across Cases 1, 2, and 3, when Law 1 is used. Meanwhile, as presented inFig. 6.17 (a), Cases 1 to 3 exhibit the same temperature profile at the âFocal point 2â on finishedsurfaces, indicating consistency in cooling rates of the surface materials across three cases duringcooling down. Consequently, the evolution of tensile stress follows the same pattern across Cases 1to 3 after the cutting tool moves away.Analogue to Cases 1 to 3, when Law 2 is employed while dg x is 10 µm, Cases 4 to 6 presentthe same average local normal (around 35.8 N/mm) and shear (about 4.4 N/mm) contact forces,as illustrated in Fig. 6.18 (e)-(f). Due to the thermo-mechanical loads applied by the cutting tool,as shown in Fig. 6.17 (d), Ïxx firstly transitions into compressive states with the advancement ofthe cutting tool towards âFocal point 2â. Then, as the tool edge begins to separate from âFocalpoint 2â, the resulting tensile strain (caused by tool traction on newly generated surfaces) reducesthe magnitude of the existing compressive stress. However, due to the consideration of stress stateeffects on materialâs flow stress (quantified by η and θ¯ in Law 2), the maximum tensile strains onfinished surfaces generated by tool traction are closely related to the initial residual stress states,as shown in Fig. 6.18 (f). Specifically, the peak tensile strains produced in the post-machiningof compressive-dominant regions (Case 4) are the least, less than those produced in the post-machining of zero-stress regions (Case 5), while post-machining of tensile-dominant regions (Case6) generate the highest tensile strains. Therefore, once the tool fully separates from âFocal point2â, the remaining compressive stress (Ïxx) is the highest for Case 4 and the lowest for Case 6.As the cutting tool moves further away, the finished surfaces begin to cool down. The generatedtensile stresses during cooling down, caused by the restricted thermal contraction limited by thesurrounding bulk material, continue to counteract the remaining compressive stress (Ïxx) on thenewly finished surfaces. Analogue to Cases 1 to 3, after the tool passes the âFocal point 2â, Cases4-6 share the same temperature evolution curves, indicating the same amount of counterbalancingtensile stress produced on the finished surfaces during cooling down. Consequently, Case 4 has thehighest magnitude of compressive residual stress on finished surfaces, while Case 6 is the lowest.It should also be noted that under Law 2, Cases 7/8/9 mirror the same thermo-mechanical load-ing processes as Cases 4/5/6. As a result, post-machining of free-of-stress regions (Case 8) resultsin compressive residual stress on finished surfaces that is higher than those generated in Case 9(post-machining of tensile-dominant areas), but lower than Case 7 (post-machining of compressive-dominant areas). Nevertheless, as the dg x increases from 10 µm (Cases 4 to 6) to 100 µm (Cases7-9), due to the decrease in materialâs yield strength, the average normal and shear contact forcesacted by the cutting tool reduced by 6.2 N/mm and 1.0 N/mm, respectively. Therefore, the thermo-mechanical loads exerted on the finished surface to form the peak compressive stress in Cases 7-9are lower than Cases 4-6. Consequently, the final compressive residual stresses on finished surfacesin Cases 7 to 9 are lower than in Cases 4 to 6.1566.4. Quantitative Analyses of Surface Residual Stress after Post-Machining DEDFigure 6.15: Residual stress distribution along the cutting velocity (x) direction: Subfigures (a)-(c) show three maps of initial residual stress; Subfigures (d)-(l) present the corresponding finalresidual stress patterns after completing the 9 post-machining simulation cases.1576.4. Quantitative Analyses of Surface Residual Stress after Post-Machining DEDFigure 6.16: Residual stress profiles before and after post-machining simulations: Subfigure (a)presents the initial residual stress profile (Initial profile 1) and the corresponding final residualstress profiles under Cases 1, 4, and 7; Subfigure (b) plots the initial residual stress profile (Initialprofile 2) and the resulting final residual stress profiles under Cases 2, 5, and 8; Subfigure (c) detailsthe initial residual stress profile (Initial profile 3) and final curves under Cases 3, 6, and 9.1586.4. Quantitative Analyses of Surface Residual Stress after Post-Machining DEDFigure 6.17: Evolution of temperature (TEMP), stress along the cutting velocity (x) direction(Ïxx), state of stress parameters (η and θ¯), and plastic strain along the cutting velocity (x) direction(PExx) at a selected point on final surfaces in post-machining simulations: Subplots (a) and (d)represent TEMP and Ïxx curves for Cases 1-3 and Cases 4-6, respectively; Subplots (b) and (e)depict the traces of η and θ¯ for Cases 1-3 and Cases 4-6, respectively; Subplots (c) and (f) showthe PExx profiles for Cases 1-3 and Cases 4-6, respectively.1596.4. Quantitative Analyses of Surface Residual Stress after Post-Machining DEDFigure 6.18: Distribution and evolution of contact forces on the cutting tool: Subfigure (a)presents the distribution of shear contact force at t = 1.35 ms under Case 4; Subfigures (b) to (j)plot the evolution of the sum of local normal and shear contact force for Cases 1 to 9, respectively.1606.5. SummaryThe above analyses suggest that no matter the initial residual stress states and grain sizevalues, Law 1 yields a consistent tensile residual stress on the final surface. This is caused by theuniform tensile strain generated by the tool traction on the newly formed surface and the restrictedthermal contraction of the surface material once the tool moves away. Conversely, Law 2 capturesa reduced compressive residual stress on the final surface. This reduction occurs as the initialresidual stress state transitions from compressive-dominant to tensile-dominant, because the initialcompressive residual stress limits the peak tensile strain caused by the tool traction, or as the initialgrain size values increase, since the larger grain size values result in less thermo-mechanical loadsacted by the cutting tool on the final surface. Therefore, Law 2 is recommended for accuratelycharacterizing the final residual stress after post-machining of DED components, particularly forrepairing critical parts and no further post-treatments are involved. However, considering that post-machining processes may introduce micro-defects on the final surface, technologies like polishingor etching are required to remove the work-hardening layer caused by machining. Meanwhile,under certain post-machining parameters, the resulting compressive residual stress layer may beinsufficient for specific applications, and additional surface treatments like shot peening and lasershock processing are necessary to thicken its depth. In such cases, predicting the post-machiningpenetration depth is more critical than quantifying the surface residual stress states, and both Law1 and Law 2 are qualified.6.5 SummaryThis chapter develops a numerical model that explains the formation mechanism of surface residualstress after the machining of IN718 fabricated by DED processes and quantifies the cumulativeeffects of initial residual stress states and grain size values on cutting forces and final surfaceresidual stress distribution. The key findings are listed below:(1) During post-machining processes, the applied thermo-mechanical loads by the cutting toolreshape the original residual stress within a limited penetration depth. For DED-built single-trackIN718 samples, XRD measurements suggest that the finished surface residual stress starts to followthe initial profiles at about 230 µm below the post-machined surface. The numerical modellingprocedure quantifies this depth as around 194 µm.(2) Variations in the initial residual stress within the chip formation region have a negligibleeffect on cutting forces. This is because, in the post-machining processes, the tool advancementcreates a localized stress concentration region dominated by compressive stress that overrides pre-existing residual stress fields ahead of the cutting tool. This consistency creates a uniform thermo-mechanical response of IN718 near the cutting edge.(3) Final surface residual stress varies with the initial residual stress states. When the initialgrain size values are fixed, post-machining of regions dominated by initial compressive residualstress features a larger compressive residual stress on the final surface compared to post-machiningof originally tensile-dominated regions. This is due to the initial compressive residual stress that1616.5. Summarylimits the peak tensile strains generated by the tool traction during post-machining, leading to ahigher net compressive stress left on the final surface.(4) An increase in the average grain size along the cutting velocity direction reduces the finalcompressive residual stress. This reduction is attributed to the lower initial yield strength of thecoarser grain regions, which results in a decrease in the average normal and shear contact forcesexerted by the cutting tool during post-machining, compared to finer grain regions.162Chapter 7Conclusions and Future ResearchDirectionsThe residual stresses in the manufactured metallic components can lead to geometric distortionsand undesired surface integrity, compromise dimensional tolerances, and reduce the service life.This thesis investigates the formation mechanisms of residual stresses in both DED and machiningprocesses, which are commonly used for remanufacturing and repair applications. A series ofanalytical and numerical models are developed and validated to bridge the knowledge gaps in (1)bulk residual stress formation in single-material and functionally graded DED processes, (2) surfaceresidual stress generation in wet machining and high-speed machining with serrated chips, and (3)the interaction between the two residual stresses in post-machining processes. The following detailsthe key findings and contributions of this thesis.To capture the thermo-mechanical response in single-material DED processes, a new analyticalmodel is developed, which has the novelty of featuring incremental material addition with the heatsource movement. The developed analytical model improves computational efficiency without sac-rificing prediction accuracy, compared to existing FEM-based models. Using the validated model,two sets of DED parameters (linear energy density parameters, defined as the ratio between heatsource power and traverse speed, and process planning parameters, including idle time and deposi-tion patterns) are examined. Simulation results highlight that in the fabrication of single-materialthin-wall structures through DED, reducing linear energy density avoids excessive heat accumula-tion and promotes uniform thermal contraction of the deposited surface, leading to reduced tensileresidual stress and slight geometric distortion. However, when the linear energy density remainsconstant, lowering heat source power and traverse speed reduces tensile residual stress and verticaldistortion. This is because these deposition conditions introduce fewer energy inputs and enhanceheat dissipation, thereby lowering thermal gradients and thermal cycles experienced by the de-posited material. Meanwhile, regardless of the deposition pattern (unidirectional or bidirectional),increasing idle time reduces thermal gradients and built-up thermal stresses in the deposit-substratesystem, thereby lowering tensile residual stress and geometric distortion. However, excessively longidle times do not proportionally decrease tensile residual stress and geometric distortion.The single-material analytical model is extended to focus on DED-fabricated FGM structures.The enhanced model accounts for the actual material composition variation within FGM structuresto simulate the bulk residual stress distribution. Residual stresses measured from DED-producedSS316/IN718 FGM thin-wall structures are used to validate the enhanced model, and NRMSE is163Chapter 7. Conclusions and Future Research Directionsemployed to quantify the deviation between measured and simulated residual stress profiles. Resultsshow that designed composition paths yield an NRMSE of 22.59%, whereas employing actual Fe-,Ni-, or Ti-based weight ratios reduces the NRMSE to 21.55â21.56%. Conversely, Nb-based predic-tions exhibit a higher NRMSE of 26.42%, due to the smaller proportion of Nb in the SS316/IN718structure. The validated model is further used to examine bulk residual stress distribution intwo types of FGM thin-wall structures featured by tilted material transition regions: vertical- andhorizontal-dominant walls. Results demonstrate that maximum tensile residual stresses are pro-portionally related to the material gradient angle in vertical-dominant walls while increasing thematerial gradient angle in horizontal-dominant walls results in more compressive residual stressesat the FGMâs top surface.A fully coupled thermo-mechanical model based on FEM is developed to simulate the surfaceresidual stress in machining processes with coolant application. The model uses the CEL tech-nique to simulate tool/workpiece/fluid interactions in material removal processes, followed by theLagrangian-based technique for cooling down to improve computational efficiency. Meanwhile, thedetermination of cutting fluid velocity losses accounts for fluid-structure interactions. This modelis experimentally validated by comparing the simulated cutting forces and surface residual stressprofiles with measurements in the orthogonal cutting of Waspaloy under both dry and wet cuttingconditions. Simulation results indicate that heat generated at the tool edge remains unaffected bycutting fluid because the fluid is not able to reach the tool-workpiece interface. However, the use ofcutting fluid yields a substantial decrease in surface temperature at around 60 °C. The reduction isattributed to the occurrence of forced heat convection at the fluid-structure interface, which enablesthe removal of more heat from the machined surface compared to dry cutting. Consequently, in wetmachining conditions, the maximum workpiece temperature occurs beneath the machined surface.Furthermore, the use of cutting fluid mitigates thermal gradients between the surface and bulkmaterials after the tool is removed. The reduction in thermal gradients leads to a more uniformcontraction of the surface material during cooling down. Therefore, a decrease in surface tensileresidual stress is observed when machining with cutting fluid compared to dry cutting.The developed FEM-based machining model is also employed to explain the fluctuation ofsurface residual stress associated with serrated chip formation in the high-speed machining ofWaspaloy. The model is further validated by comparing the simulated chip morphologies andmachined surface profiles with measurements. Analyses of the normal/shear contact forces atthe tool-workpiece interface demonstrate that during continuous chip formation conditions, thestagnation point remains stationary on the cutting tool with the absence of shear localization.This results in a relatively flat surface, which is characterized by a uniformly distributed residualstress on the finished surface. However, at a higher cutting velocity, the generation of serratedchips, caused by periodic shear band formation, induces cyclical movement of the stagnation pointon the cutting tool. This movement leads to variations in the volume of workpiece material beingplowed by the cutting tool, generating waviness on the finished surface due to periodicity in theelastic recovery of plowed materials after the cutting tool moves away. Moreover, the cyclic motion164Chapter 7. Conclusions and Future Research Directionsof the stagnation point induces time-varying thermo-mechanical loads applied by the cutting toolon the machined surface. Therefore, surface residual stress fluctuates along the cutting velocitydirection with the same frequency as chip segmentation.To predict the final residual stress following the machining of DED-fabricated structures, theFEM-based machining model integrates two critical DED-induced factorsâbulk residual stressstates and microstructural characteristicsâas initial conditions. In the orthogonal cutting sim-ulation of DED-built IN718 single-track samples, the developed model effectively estimates thepenetration depth of post-machining into the initial residual stress distribution, which aligns withresidual stress measurements. Based on this validated model, the cumulative effects of initial resid-ual stress states and grain size values on cutting forces and final surface residual stress profiles aresystematically quantified. Simulation results show that variations in the initial residual stress inthe chip formation region have a negligible effect on cutting forces. This is because the localizedcompressive stress generated due to the tool advancement overrides any pre-existing residual stressstates, creating a consistent thermo-mechanical response near the tool edge. However, the finalsurface residual stress varies with the initial residual stress states. Specifically, due to the initialcompressive residual stress limiting the peak tensile strain caused by tool traction, machining re-gions with initial compressive stress leave higher net compressive residual stress on the final surfacethan regions with initial tensile stress. More importantly, compared to regions with finer grain sizevalues, machining regions with coarser initial grain size values generate less compressive residualstress on the final surface. This is attributed to the lower initial yield strength of the coarser grainregions, which results in less average normal and shear contact forces exerted by the cutting toolto form the final surface residual stress.Future possible research directions to model the manufacturing-induced residual stresses arelisted as follows:(1) Modelling of Bulk Residual Stress in AM Structures: First, future studies are suggestedto integrate the effects of porosity and phase transformation into the prediction of DED-inducedbulk residual stress. Meanwhile, the current DED model could be expanded to account for heatlosses and thermal stress generation in the transverse direction, enabling predictions of residualstress and distortion in 3D spaces for more complex geometries. Furthermore, the model couldbe adapted to powder bed fusion (PBF) processes to predict bulk residual stress in componentswith intricate geometries, such as overhangs, internal cavities, and lattice structures. Therefore, ageneralized bulk residual stress modelling framework for metal AM processes could be formulatedthrough comparative analyses of DED and PBF.(2) Prediction of Surface Residual Stress in Machining Processes: First, based on the developedwet machining model, optimizing cutting fluid parameters could be explored to achieve the desiredsurface integrity. Next, for serrated chip conditions, variations in peak compressive residual stresslocations and their penetration depth could be quantified, along with their influences on fatiguelife. Besides, actual tool wear morphologies are suggested to integrate into the machining model,accounting for the actual contact area between the tool and the machined surface to investigate their165Chapter 7. Conclusions and Future Research Directionseffect on surface residual stress generation. Eventually, the developed FEM-based 2D machiningmodel could be extended to assess the surface residual stress caused by 3D general machiningprocesses, such as turning and milling processes, for more practical applications.(3) Estimation of Surface Residual Stress Following Post-Treatments of AM Structures: First,the developed post-machining model could be updated by setting the post-machining-induced resid-ual stress as initial conditions to predict the ultimate surface residual stress after post-treatmenttechniques, such as shot peening or laser-assisted surface treatments, to increase the compres-sive residual stress layerâs thickness. Moreover, differences in final surface residual stress afterpost-machining of DED-produced single-material, FGM, and multi-material structures are highlysuggested to quantify, to engineer the residual stress profiles by customizing material composi-tion ratios within DED-fabricated FGM and multi-material structures. 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Specifically, the cell size in the heat source traverse direction depends onthe dimensions of the heat source beam, with a minimum ratio of 5 between the heat source beamand the cell size. Moreover, the cell size in the build direction should be at least 4 times smallerthan the layer thickness.Once the cell size âl is determined, the selection of the time increment ât is based on thefollowing criterion to ensure calculation stability:ât â¤ min[Ï(T ) · Cp(T )4 · k(T ) · (âl)2]. (A.1)The above equation refers to the convergence condition by Courant-Friedrichs-Lewy, which wasoriginally summarized based on the temperature-independent material properties when solving par-tial differential heat transfer equations using the finite difference method. Since the current modelemploys temperature-dependent material properties, the selected time increment should be theminimum value, satisfying all the convergence conditions in the whole temperature range of thedeposit-substrate system during the deposition process. Therefore, it is necessary to identify themaximum temperature (Tmax) in the deposit-substrate system during deposition. The determina-tion of Tmax requires two steps. At first, Tmax is estimated based on the minimum time increment(named âoriginal time incrementâ) calculated by the temperature-dependent material propertieswith the temperature ranging from the ambient temperature to the melting point. If the simulationresult converges, the obtained Tmax is used to correct the âoriginal time incrementâ, to make thesimulation more efficient. The corrected value is the final time increment used in the simulation.Otherwise, the âoriginal time incrementâ needs to reduce (usually by 0.1 ms according to the de-position conditions), and the above simulations need to be repeated until the simulation resultconverges.184Appendix BThe Necessity to Include SurfaceNormal Tractions and HydrostaticStresses in DED-Induced BulkResidual Stress CalculationAs shown in Fig. B.1, the red points represent the longitudinal residual stress predicted by a3D FEM-based model [41], the blue solid curve shows the simulation results by the proposedmodel (model 1) considering all three stress components, and the green long-dashed line showsthe prediction by the model (model 2) only considering the body forces in the thermal stresscalculation. The above calculations share the same deposition conditions as the residual stressvalidation (Section 2.2.2). The results suggest that when thermal stresses caused by the normaltraction on the top surface and the hydrostatic stress are neglected, the peak tensile residual stresson the substrate remains the same as the results predicted by model 1, which considers all threestress components (body forces, surface normal tractions, hydrostatic stresses) into the thermalstress calculation. However, neglecting the two stress components results in less tensile residualstress at other deposit locations and more compressive residual stress on the substrate. As a result,the maximum vertical distortion is underestimated, as shown in Fig. B.2, which compares themeasured distortion profile [41] (shown as red points), the predicted distortion curve (the blue solidline) based on model 1, and the distortion profile (the green long-dashed line) based on model 2.It can be seen that the residual stress profile from model 1 yields the maximum prediction errorof 17.7% for the peak vertical distortion. However, this prediction error increases to 57.1% whenthe surface normal tractions and hydrostatic stresses are neglected. Therefore, it is recommendedto include the normal traction acting on the top surface and the hydrostatic stress in the thermalstress calculation.185Figure B.1: Comparison of longitudinal residual stress in thin-wall DED-built structures betweenFEM model [41], the prediction from model 1 (considering all three stress components), and theprediction from model 2 (considering only body forces).Figure B.2: Comparison of measured [41] distortion curves for a DED-built thin-wall structure,the prediction from model 1 (considering all three stress components) and the prediction frommodel 2 (considering only body forces).186Appendix CEstimating Relaxed Bulk ResidualStress in DED-Fabricated FGMStructures Due to Material Removalthrough WEDM and ElectropolishingA 3D FEM-based numerical modelling procedure is developed to calculate the relaxed residualstress during sample preparation prior to XRD measurements for DED-built FGM structures. Asdemonstrated in Fig. C.1, four steps are involved in obtaining the final residual stress profiles afterWEDM and electropolishing: FGM fabrication (Step 1), cooling down (Step 2), material removal(Step 3), and stress rebalancing (Step 4). All these steps are conducted using a commercial FEMsolver ABAQUS. Specifically, in Step 1, to capture the material response during the fabrication ofFGM structures, apart from assigning material properties and meshing strategies, the key pointlies in how to accurately define element activation in relation to the moving heat input, to mimicthe incremental material addition with the movement of heat source in the DED process. UsingABAQUS, oftentimes, the âelement death and birthâ technique coupled with a user-subroutineDFLUX is used to simulate the thermo-mechanical response in DED processes. However, this ap-proach involves manually assigning the path of element activation according to the element positionduring deposition. As a result, the pre-processing stage is time-consuming, especially for large-scaleindustrial-level metallic components. As an alternative, this study employs event series to achieveautomatic element activation during the DED simulation. This automatic activation algorithm re-quires the definition of event series data as a function of both element position and deposition time[148]. In the DED simulation, the user-subroutine UEPACTIVATIONVOL is used to interpret theevent data and incrementally activate elements, while the user-subroutine UMDFLUX is employedto progressively apply the heat input to the FGM-substrate system based on the defined seriesdata. As reported by Song et al. [148], the âUEPACTIVATIONVOL+ UMDFLUXâ approach notonly reduced the pre-processing time but also saved the total computational time, compared tothe traditional âelement birth/death + DFLUXâ method. Once all elements are activated, Step 2starts. This step is designed to unlock the thermal constraints within each element by cooling theFGM-substrate system to room temperature. After Step 2, the remaining plastic stresses in theFGM structure are identified as the DED-induced residual stress, referred to as the original resid-ual stress profile. Then, to simulate the relaxation of DED-induced residual stress during sample187preparation, the âmodel changeâ method is employed in Step 3, where all elements correspondingto the WEDM + electropolishing regions are deactivated. This removal of material alters the initialequilibrium in the original FGM structure. As a result, the remaining FGM structure will undergofurther deformation until a new equilibration condition is reached. Step 4 simulates this stressrebalancing process within the remaining FGM structure. At the end of Step 4, the rebalancedplastic stresses are identified as the DED-induced residual stress after relaxation, referred to asthe redistributed residual stress profile. By calculating the stress difference between the originaland the redistributed residual stress profiles, the residual stress relaxation caused by WEDM +electropolishing can be estimated. While there are several analytical approaches in the literature[52, 185] to estimate the redistributed residual stress profiles after material removal, our FEM-basedmodel provides a more accurate way. This is because, in this study, the WEDM + electropolish-ing process removes the material in the transverse direction. Nevertheless, the 2D analytical andpractical model only calculates the longitudinal residual stress distribution in the mid-longitudinalcross-section of the FGM structure, without quantifying the longitudinal residual stress variationin the transverse direction. As a result, if we use the existing analytical approaches [52, 185] forquantifying the relaxed residual stress, there will be no inputs to calculate the equivalent forces andmoments for satisfying a new force and moment equilibration within the remaining FGM structure.Therefore, considering the feasibility to perform both material addition and subtraction processesnumerically, we developed a 3D FEM-based modelling procedure to first generate the originallyDED-induced residual stress for the whole FGM structure and then quantify the relaxed residualstress in the remaining FGM structures after WEDM + electropolishing. Based on this quantifi-cation, the XRD measurements can be corrected by removing the relaxed stress caused by WEDM+ electropolishing, which will be used for model validation.Following the FEM-based modelling procedures described above, Fig. C.2 (a) and (b) presentthe longitudinal residual stress distribution before and after stress redistribution in the remainingFGM structure, respectively. The material properties and deposition conditions are the same asthe analytical model presented in Section 3.3, under the designed material transitional path. Inthe simulation, the âoriginal FGM sampleâ shares 22,400 8-node linear brick elements (C3D8R),while the âremaining FGM sampleâ has 19,264 C3D8R elements. The 3D model, under the samecomputational resources as our analytical model, required a total of 17.3 hours to complete thesimulation of the FGM fabrication process (Step 1, as shown in Fig. C.1). Fig. C.2 (c) compares theresidual stress profiles before and after stress relaxation at a selected path in the mid-longitudinalcross-section of the FGM structure. The results suggest that the average relaxed residual stresscaused by the combined effects of WEDM and electropolishing accounts for 15.9% of the originalDED-induced residual stress. Given the necessity to perform WEDM + electropolishing beforeXRD measurements in this study, we have also included the relaxed residual stress as part ofmeasurement variations shown in Fig. 3.8.188Figure C.1: Initial geometry and boundary conditions of the proposed 2D FEM-based modellingprocedure to estimate the relaxed residual stresses.Figure C.2: Comparison of residual stress profiles before and after stress relaxation due to spec-imen removal by WEDM.189Appendix DQuantifying Relaxed Residual Stresson Machined Surfaces Caused bySeparating Specimens Using WEDMA 2D FEM-based numerical modelling procedure is developed to calculate the amount of residualstress relaxation caused by removing the Waspaloy specimen from the original workpiece usingWEDM for XRD characterization. The model is formulated by the Lagrangian method usingABAQUS/Implicit mode. As shown in Fig. D.1, based on the developed model, two residual stressprofiles can be obtained: before and after the specimen removal. The original residual stress pro-file refers to the residual stress distribution before specimen removal. This profile is obtained byloading the equivalent mechanical loads and heat sources on the machined surface of the âoriginalworkpieceâ as the cutting process, then cooling the workpiece to room temperature. Accordingto Ulutan et al. [159], the equivalent mechanical loads (in the normal and shear directions) andheat sources (in the primary and secondary zones) can be calculated based on the tool geometriesand measured cutting forces (tangential and feed forces). In the loading process, the equivalentmechanical-thermal load moves along the cutting velocity direction with a velocity of Vc. The move-ment of the thermal-mechanical loads is achieved by user-subroutines: DLOAD (mechanical loadsin the normal direction), UTRACLOAD (mechanical loads in the shear direction), and DFLUX(thermal loads). After the loading process is completed and the workpiece cools down to roomtemperature, the plastic stress and strain at the material points in the focal area are extracted andintegrated into the âremoved sampleâ as shown in Fig. D.1. The âremoved sampleâ shares the samedimensions as the specimens cut by WEDM in the cross-section along the cutting velocity direction.By removing the boundary conditions at the side and bottom surfaces, the âremoved sampleâ hasstress relaxation by redistributing the residual stress to satisfy the equilibrium condition. Finally,the average relaxed residual stress is calculated by averaging the absolute stress difference of theresidual stress profiles before and after specimen removal along the vertical direction.Fig. D.2 compares the residual stress profiles before and after the stress relaxation caused by thespecimen removal. The original residual stress is generated by loading the equivalent mechanicalloads. The equivalent mechanical loads are calculated based on the measured cutting forces underVc = 86 m/min, H = 0.12 mm, and wet cutting. In the calculation, the âoriginal workpieceâshares 196,800 4-node thermally coupled Lagrangian quadrilateral (CPE4RT) elements, while theâremoved sampleâ has a total of 84,072 CPE4RT elements. Under the given cutting condition, the190result suggests that around 42.3% of the compressive residual stress at the top of the machinedsurface has been relaxed due to specimen removal, while the average relaxed residual stress isaround 15.9%, which has been included in error bars of Fig. 4.14.Figure D.1: Initial geometry and boundary conditions of the proposed 2D FEM-based modellingprocedure to estimate the relaxed residual stress.Figure D.2: Comparison of residual stress profiles before and after stress relaxation due to spec-imen removal by WEDM (Vc = 86 m/min, H = 0.12 mm, and wet cutting).191Appendix ECharacterization of Stress States byStress Triaxiality and Lode ParameterStress triaxiality and Lode parameter are used to feature the initial and in-process stress statesin DED-built sturctures during post-machining, whose definition is based on the following threeinvariants [16]:pm = âÏm = â13(Ï1 + Ï2 + Ï3), (E.1)q =â32S : S =â12[(Ï1 â Ï2)2 + (Ï2 â Ï3)2 + (Ï3 â Ï1)2], (E.2)r =(92S : S : S)1/3=[272(Ï1 â Ïm)(Ï2 â Ïm)(Ï3 â Ïm)]1/3, (E.3)where pm is the hydrostatic stress; q is von Mises equivalent stress; r is the third invariant ofdeviatoric stress; Ï1, Ï2, and Ï3 denote the three principal stresses; Ïm is mean principal stress;and S refer to the deviatoric stress tensor.Taking an initial residual stress vector on the space of principle stresses as an example, theresidual stress vector can be decomposed into two parts: the deviatoric part (on the deviatoricplane, where Ï1 + Ï2 + Ï2 = 0) and the hydrostatic part (on the hydrostatic axis, where Ï1 =Ï2 = Ï3). The ratio between the hydrostatic part and the deviatoric part is defined as the stresstriaxiality (η), which is expressed as [16]:η = âpmq, (E.4)η is associated with an angle between the initial residual stress vector and the deviatoric plane,which is responsible for the size of the initial yield surface.In addition, the Lode angle (θ) is also defined on the deviatoric plane, which refers to thesmallest angle between the line of pure shear and the projection of the initial stress vector on thedeviatoric plane. The shape of the yield surface is associated with the value of θ, whose calculationis related to the normalized third deviatoric stress invariant (ξthrid), expressed as [16]:ξthird =(rq)3= cos(3θ). (E.5)192Since the range of θ is between 0 and Ï/3, the range of ξthird is between -1 to 1. Furthermore,θ can be normalized by [16]:θ¯ = 1â 6Î¸Ï= 1â 2ÏÂ· arccos(ξthird), (E.6)where θ¯ is the Lode parameter, whose range is between -1 and 1.193
Thesis/Dissertation
eng
Mechanical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Jin, Xiaoliang
Clare, Adam T.
Modeling of residual stresses in combined additive manufacturing and machining processes
Text