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Numerical modeling of orthogonal cutting of carbon fibre reinforced polymer composites Garekani, Amir Hossein Afrasiabi 2016

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 NUMERICAL MODELING OF ORTHOGONAL CUTTING OF CARBON FIBRE REINFORCED POLYMER COMPOSITES   by  Amir Hossein Afrasiabi Garekani  B.ASc in Civil Engineering, University of British Columbia 2013   A THESIS SUBMITTED IN PARTIAL FULLFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCES  in  THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDEIS  (Civil Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA  (Vancouver)   July 2016  ©Amir Hossein Afrasiabi Garekani, 2016    ii  Abstract  The focus of this study is on the orthogonal cutting of fibre reinforced composites. A thorough review of the literature is presented in both experimental and numerical work that has thus far been conducted. It is found that in orthogonal cutting of composites, cutting forces, chip formation mechanism and the extent of damage below the cutting plane are highly dependent on the fibre orientation. With fibre orientation increasing from 0° to 90°, cutting forces tend to increase and chips become more dust-like. Two modeling approaches are most commonly adopted for the prediction of force and chip formation, namely the micro-mechanical and the macro-mechanical approach; in the former, the fibre and the matrix are modeled as separate phases and their interface is defined by a traction-separation law. In the latter, the composite is represented by an anisotropic equivalent homogeneous material. It is shown that the numerical predictions are in good agreement with experimental results. With ABAQUS being the most commonly used tool for modeling of composite orthogonal cutting, a two-dimensional macro-mechanical model for cutting of CFRP was created in ABAQUS using Hashin’s damage model. The cutting forces were predicted for fibre orientations of 0°, 30°, 45°, 60°, 90° and 135°. Good prediction of cutting forces with the available experimental data in the literature was obtained. The model however fails to predict the complete chip formation mechanism due to limitations in the material model. A similar model was developed in LS-Dyna using MAT_054 for composites. The cutting forces were found to be sensitive to the damage input parameters, mainly the strain-to-failure of the elements. However, the complete chip formation and chip release was captured in this model which correlated well with the experimental observations.  It is suggested that for improved modeling capability, better understanding of the damage behaviour of FRP, especially at the micro-scale is needed. Also, the existing work is found to be limited to orthogonal cutting and drilling separately; it is however worth looking into both items in one study and linking the two processes through a geometric transformation  which is the established framework in metal machining.   iii  Preface This thesis is the work of Amir H. A.Garekani which was carried out under the supervision of Dr. Reza Vaziri at the Composite Research Network at the University of British Columbia. The research topic was identified by Dr. Reza Vaziri (Supervisor) and Dr. Yusuf Altintas from the Mechanical Engineering department at the University of British Columbia.  The literature review conducted is the work conducted by the author as well as the numerical models developed in this study. The dimensions and boundary conditions implemented in the numerical models were adopted from models previously developed in the literature. The sensitivity analysis conducted to identify the variation in the results to the input parameters were conducted by Amir H. A. Garekani. The conclusions drawn from this study and the suggested future work are based on the thorough literature review of publications in the field of Composite Machining from 1983 to 2015 and the results from the numerical analysis conducted.             iv  Table of Contents Abstract ........................................................................................................................................... ii Preface............................................................................................................................................ iii Table of Contents ........................................................................................................................... iv List of Tables ............................................................................................................................... viii List of Figures ................................................................................................................................ ix Acknowledgements ....................................................................................................................... xii  Overview and Research Objectives .............................................................................. 1 Chapter 11.1 Introduction ...................................................................................................................... 1 1.2 Composite Machining ...................................................................................................... 1 1.3 Orthogonal Cutting in Metals ........................................................................................... 2 1.4 Orthogonal Cutting in Composites ................................................................................... 5 1.5 Objective and Goals ......................................................................................................... 6 1.6 Thesis Layout ................................................................................................................... 6  Orthogonal Cutting of Composite: A Review of Experimental Studies ...................... 7 Chapter 22.1 Introduction ...................................................................................................................... 7 2.2 Uni-Directional Composites ............................................................................................. 8 2.2.1 0° Fibre Orientation ................................................................................................... 8 2.2.2 90° Fibre Orientation ............................................................................................... 11 2.2.3 45° Fibre Orientation ............................................................................................... 12 2.2.4 135° and Greater Fibre Orientations ....................................................................... 13 2.3 Chip Generation ............................................................................................................. 14 2.4 Bouncing Back Phenomena ........................................................................................... 15 2.5 Forces in The Orthogonal Cutting of Composites ......................................................... 16 2.5.1 Cutting Forces ......................................................................................................... 16 2.5.2 Thrust Forces .......................................................................................................... 21 v  2.5.3 Influence of Tool Geometry.................................................................................... 24 2.6 Multidirectional Composites .......................................................................................... 27 2.6.1 Chip Formation ....................................................................................................... 27 2.6.2 Cutting Forces ......................................................................................................... 28 2.7 Summary ........................................................................................................................ 28  Orthogonal Cutting of Composites: A Review of Numerical Studies........................ 31 Chapter 33.1 Introduction .................................................................................................................... 31 3.2 Modeling Approach........................................................................................................ 32 3.2.1 Macro-Mechanical Approach ................................................................................. 32 3.2.2 Micro-Mechanical Approach .................................................................................. 32 3.2.3 Input Parameters ..................................................................................................... 33 3.3 Macro-Mechanical Models ............................................................................................ 34 3.3.1 Two-Dimensional ................................................................................................... 34 3.3.2 Three-Dimensional ................................................................................................. 37 3.4 Micro-Mechanical Models ............................................................................................. 40 3.5 Summary ........................................................................................................................ 45  Composite Material Models in ABAQUS and LS-Dyna ........................................... 47 Chapter 44.1 Introduction .................................................................................................................... 47 4.2 Hashin’s Damage Model in ABAQUS .......................................................................... 47 4.2.1 Damage Initiation.................................................................................................... 48 4.2.2 Damage Evolution .................................................................................................. 49 4.2.3 Element Erosion in ABAQUS Explicit................................................................... 51 4.2.4 Single Element Examples ....................................................................................... 51 4.3 LS-Dyna MAT_054 Enhanced Composite Damage ...................................................... 58 4.3.1 Strength Parameter .................................................................................................. 58 vi  4.3.2 Strain-to-Failure Parameter ..................................................................................... 58 4.3.3 Single Element Runs in LS-Dyna ........................................................................... 59  Orthogonal Cutting of Metals: Numerical Modeling ................................................. 62 Chapter 55.1 Introduction .................................................................................................................... 62 5.2 Orthogonal Cutting of Metals in ABAQUS ................................................................... 63 5.2.1 Johnson-Cook Constitutive Model ......................................................................... 63 5.2.2 Finite Element Model ............................................................................................. 64 5.2.3 ABAQUS Results ................................................................................................... 67 5.3 Orthogonal Cutting of Metals in LS-Dyna ..................................................................... 69 5.3.1 Johnson-Cook Constitutive model .......................................................................... 69 5.3.2 Model Description in LS-Dyna ............................................................................... 70 5.3.3 LS-Dyna Results ..................................................................................................... 71 5.3.4 Sensitivity Analysis of Results to the Strain-to-Failure Parameter ........................ 73  Orthogonal Cutting of Composites: Numerical Simulation ....................................... 77 Chapter 66.1 Introduction .................................................................................................................... 77 6.2 Orthogonal Cutting of Composites in ABAQUS ........................................................... 77 6.2.1 Mesh Size and Element Type ................................................................................. 79 6.2.2 Hashin’s Damage Model: Input Parameters ........................................................... 81 6.2.3 Boundary Conditions and Contact in ABAQUS .................................................... 82 6.3 Orthogonal Cutting of Composites in LS-Dyna ............................................................. 83 6.3.1 Mesh Size and Element Type ................................................................................. 84 6.3.2 MAT_054 Damage Model: Input Parameters ......................................................... 85 6.3.3 Boundary Condition and Contact in LS-Dyna ........................................................ 86  Results and Discussion ............................................................................................... 87 Chapter 77.1 Cutting Force Plots From ABAQUS .............................................................................. 87 vii  7.2 Cutting Force Plots From LS-Dyna ............................................................................... 98  Conclusions and Future Work .................................................................................. 103 Chapter 8Bibliography ............................................................................................................................... 104                        viii  List of Tables  Table  2-1_ List of experiments on orthogonal cutting of composites .......................................... 30 Table  3-1_List of numerical models on orthogonal cutting of composites .................................. 46 Table  4-1_Mechanical properties of AS4/3501-6 CFRP composite ............................................ 53 Table  4-2_Expected outcome of the single element test for fibre compression mode ................. 54 Table  4-3_Summary of the steps in the single element run for testing the element in longitudinal tension and compression ............................................................................................................... 56 Table  4-4_Strain-to-failure parameters in LS-Dyna ..................................................................... 59 Table  5-1_Input parameters and material properties for Brass ..................................................... 63 Table  5-2_Geometric features of the cutting tool ......................................................................... 65 Table  5-3_Strain-to-failures values used in LS-Dyna for the orthogonal cutting of brass ........... 73 Table  6-1_Damage parameter (fracture energy density) inputs for Hashin's damage model in ABAQUS applied to CFRP composites (Santiuste, Soldani, & Miguelez, 2010) ........................ 82 Table  7-1_"Benchmark" model description in ABAQUS ............................................................ 88 Table  7-2_Cutting forces per unit width of UD CFRP obtained from the "benchmark" model (cutting depth of 0.2 mm) ............................................................................................................. 91 Table  7-3_Average cutting force values from ABAQUS model at complete chip formation with Max Degradation set to 0.98 ......................................................................................................... 96 Table  7-4_LS-Dyna strain-to-failure values used in MAT_054 ................................................... 99             ix  List of Figures Figure 1.1_Basics of Orthogonal Cutting setup .............................................................................. 3 Figure 1.2_Orthogonal cutting of metal indicating the primary (shear) zone and the force decomposition ................................................................................................................................. 4 Figure 1.3_Experimental data indicating cutting and thrust force versus cutting depth (taken from Altintas, 2012)................................................................................................................................. 4 Figure 1.4_ Transition from orthogonal cutting model to milling (adapted from Altintas, 2012) . 5 Figure 2.1_Fibre orientation in the orthogonal cutting of composites a) 0°, b) 45°, c) 90°, c) 135° (adapted from Wang et al. 1995,a) .................................................................................................. 8 Figure 2.2_Experimental test setups, a) 0°, b) 90° (taken from Koplev et al. 1983) ....................... 8 Figure 2.3_ Chip formation in 0° fibre orientation (adapted from Zitoune et al. 2005) ................. 9 Figure 2.4_Chip formation in 0° fibre orientation with a) positive and b) negative tool rake angles (adapted from Ghidossi, El Mansori, & Pierron 2006) ................................................................. 11 Figure 2.5_ Chip formation in 90° orientation (adapted from Zitoune et al. 2005) ...................... 12 Figure 2.6_Chip formation mechanism in 45° fibre orientation (adapted from Wang et al. 1995,a)....................................................................................................................................................... 13 Figure 2.7_Cutting forces versus fibre orientation in CFRP and GFRP orthogonal cutting for multiple depths of cuts (taken from Venu Gopala Rao et al. 2007,b) .......................................... 17 Figure 2.8_Cutting force versus fibre orientation for experiments conducted by Bhatnagar et al. (1995) CFRP material with cutting depth of 0.25 mm and cutting width of 2.2 mm ................... 18 Figure 2.9_Cutting force plots for multiple fibre orientations in orthogonal cutting of CFRP; cutting depth of 0.25 mm, cutting width of 4 mm, tool rake angle 10° (taken from Wang et al. 1995,a) .......................................................................................................................................... 19 Figure 2.10_Cutting force vs. Cutting depth for 0° fibre orientation (adapted from Koplev et al. 1983 and Zitonue et al. 2005) ....................................................................................................... 20 Figure 2.11_Cutting forces (N/mm) in orthogonal cutting of GFRP for multiple fibre orientations and cutting depths of 0.1, 0.2 and 0.3 mm (taken from Nayak, Bhatnagar, & Mahajan 2005,b) . 21 Figure 2.12_ Machining forces versus fibre orientation in CFRP and GFRP orthogonal cutting for cutting depth of 0.1 mm (adapted from Venu Gopala Rao, Mahajan, & Bhatnagar 2007,b) . 23 Figure 2.13_Machining force versus fibre orientation in GFRP orthogonal cutting for cutting depth of 0.2 mm adapted from Nayak et al. (2005,a) and for CFRP orthogonal cutting for cutting depth of 0.25 mm (adapted from Wang et al. 1995,a) .................................................................. 24 Figure 2.14_Cutting forces versus the rake angle of the tool for multiple fibre orientations at 0.2 mm depth of cut (adapted from Nayak et al. 2005,a) ................................................................... 26 Figure 3.1_Workpiece boundary conditions for purpose of numerical analysis .......................... 34 Figure 3.2_Primary and secondary fracture planes used in the numerical model by Arola & Ramulu (1997) .............................................................................................................................. 36 Figure 3.3_Comparison of numerical and experimental results for both cutting forces and thrust forces in orthogonal cutting of CFRP composites with a tool rake of 10° (taken from Venu Gopala Rao et al. 2008) ................................................................................................................ 38 x  Figure 3.4_Delamination damage (shown in red) in orthogonal cutting of multidirectional laminate (taken from Santiuste et al. 2011) .................................................................................. 40 Figure 3.5_Schematic view of the micro mechanics orthogonal model for orthogonal cutting of GFRP developed by Nayak et al. (2005,b) ................................................................................... 41 Figure 3.6_Micro-mechanical model for 75° fibre orientation showing separation between the fibre and the matrix phase (taken from Nayak et al. 2005,b) ....................................................... 42 Figure 3.7_Comparison of numerical and experimental cutting and thrust forces (taken from Nayak et al. 2005,b) ...................................................................................................................... 42 Figure 3.8_Transction-separation response for the interfacial cohesive elements for each opening mode .............................................................................................................................................. 43 Figure 4.1_Damage evolution stress-displacement plot for Hashin's damage model .................. 50 Figure 4.2_Single element setup used to evaluate the material response for Hashin's constitutive model............................................................................................................................................. 52 Figure 4.3_Single element setup for testing fibre compression with fracture energy of 150 kJ/m2....................................................................................................................................................... 54 Figure 4.4_Single element longitudinal compression damage initiation and evolution ............... 55 Figure 4.5_Single element run with multiple steps ...................................................................... 56 Figure 4.6_Single element results for the compression-tension-compression load element ........ 57 Figure 4.7_Stress-displacement for single element in tension...................................................... 60 Figure 4.8_Stress-displacement plot for a single element in LS-Dyna in a tension-compression cycle .............................................................................................................................................. 61 Figure 5.1_Orthogonal cutting of brass, setup in ABAQUS Explicit........................................... 65 Figure 5.2_Cutting tool in ABAQUS ........................................................................................... 66 Figure 5.3_Cutting force from orthogonal cutting simulation of brass in ABAQUS ................... 68 Figure 5.4_ Chip formation through yielding along a shear plane at 0.026 mm of workpiece displacement (shear stresses shown) in orthogonal cutting simulation of brass ........................... 69 Figure 5.5_Comparison of the brass orthogonal cutting models in ABAQUS and LS-Dyna ...... 71 Figure 5.6_Cutting forces from orthogonal cutting of brass in LS-Dyna and ABAQUS............. 72 Figure 5.7_Orthogonl cutting of brass in LS-Dyna for different strain-to-failure inputs (D1 = 150%, 300%, 450%, 750%) .......................................................................................................... 74 Figure 5.8_Effective plastic strains for cutting model with D1 = 750.0 % shown for the elements deleted in the first 0.02 mm of cutting tool displacement............................................................. 75 Figure 5.9_ Effective plastic strains for cutting model with D1 = 450.0 %  shown for the elements deleted in the first 0.02 mm of cutting tool displacement ............................................. 76 Figure 6.1_Workpiece dimensions in numerical modeling of orthogonal cutting of composites 78 Figure 6.2_Tool insert dimensions in numerical modeling of orthogonal cutting of composites 79 Figure 6.3_Workpiece setup in ABAQUS for orthogonal cutting of composites; a) partitioning of the workpiece for different mesh densities, b) meshed workpiece ............................................... 80 Figure 6.4_Material orientation angles in ABAQUS ................................................................... 83 Figure 6.5_Meshed models in ABAQUS and LS-Dyna ............................................................... 85 xi  Figure 6.6_Vectors defining material orientation angles in LS-Dyna .......................................... 86 Figure 7.1_ Cutting forces versus displacement for a) 0°, b) 45° and c) 90° fibre orientations for cutting speeds of 5.0 (green line) and 50.0 (red line) m/min ........................................................ 89 Figure 7.2_ Cutting forces versus displacement for a) 0°, b) 45° and c) 90° fibre orientations for model with Max Degradation set to 1.0 (green line) and set to 0.98 (red line) ............................ 93 Figure 7.3_Chip formation in 0° fibre orientation with Max Degradation set to 0.98 (matrix tensile damage shown) .................................................................................................................. 95 Figure 7.4_Chip formation in a) 45° and b) 90° fibre orientations ................................................ 95 Figure 7.5_Comparison of experimentally obtained cutting forces for multiple fibre orientations in CFRP orthogonal cutting .......................................................................................................... 97 Figure 7.6_Cutting force versus time for 45° UD-CFRP cutting for multiple mesh types in ABAQUS (taken from Soldani et al., 2011) ................................................................................. 98 Figure 7.7_Cutting forces versus displacement for 0°, b) 45° and c) 90° fibre orientations for models with different value of strain-to-failure .......................................................................... 100 Figure 7.8_Chip formation stages in a) 0° and b) 45° fibre orientations ..................................... 101               xii  Acknowledgements I would like to thank my supervisor Dr. Reza Vaziri for his support as a research supervisor and a teacher during the course of my study at the University of British Columbia. I would also like to thank Dr. Anoush Poursartip, the Director of the Composite Research Network at UBC for providing me with a great opportunity to interact with professionals from industry and to gain valuable academic and professional experience during my study.  I would also like to thank Alireza Forghani and Mina Shahbazi, former and current members of the CRN whom without their help and advice, this path would not have been as enjoyable as it was. A special thanks goes to all the fellow graduate students at the CRN who made my time and experience at UBC an unforgettable one.  At last, I would like to thank my parents whom without their support and kindness during all this time, this thesis would not have been possible; special thanks to you mom for your patience and tolerance during all this time and I hope to have made it up to you.  1   Overview and Research Objectives  Chapter 11.1 Introduction  Composites are now being extensively used in aerospace and automotive industries due to their superior mechanical properties such as high strength and stiffness, low weight and low thermal expansion. All of this makes composites very desirable materials especially for aerospace applications. However, in such applications, there are a number of challenges, one of which is the difficulty involved in the machining of composite components such as those made of Carbon Fibre and Glass Fibre reinforced polymer (CFRP and GFRP). Machining of these materials includes post curing processes such as trimming and drilling in order to facilitate their assembly.   1.2 Composite Machining  Machining operations in the metal industry is a rather well defined field as opposed to composites. Due to the significant difference between material properties and characteristics of composites and metals, the fundamentals of metal cutting are not directly applicable to composites (Jahromi & Bahr, 2010). This is mainly due to the anisotropic nature of the material and the difference in the chip formation and material removal process in composites. The chip formation in composites is through the brittle failure and successive rupture of the material rather than a shearing process which is the main mechanism for chip formation in metals (Lasri, Nouari, & Mansori, 2009; Venu Gopala Rao, Mahajan, & Bhatnagar, 2007,b). Another significant difference in the machining of composites is the excessive tool wear; this is mainly due to the highly abrasive nature of the fibres. It has been reported in the literature that worn out tools cause higher cutting forces and increase the level of damage in the machined part. It is found through experimental studies that the delamination that occurs during drilling increases with wearing of tool. Koplev et al. (1983) mentions short tool life and delamination as the characteristics of CFRP machining (Faraz, Biermann, & Weinert, 2009; Koplev, Lystrup, & Vorm, 1983). In metals machining, the chip is formed through the plastic deformation of the material; as the cutting tool advances, a continuous chip is formed through shearing and plastic deformation of the material. In composites however, the anisotropic and heterogeneous nature of the material makes the machining process and the induced damage highly dependent on the fibre orientation. 2  There have been both numerical and experimental studies which show the high dependency of the cutting forces on the fibre orientation. These are described in more detail in the following sections.  In order to maintain the strength and the properties of the composite part, any machining induced damages must be avoided or minimized. The most common induced damages during machining of fibre composites include delamination, fibre breakage, fibre pull out and thermal degradation and cracking of the matrix. Excessive tool wear is also known to be a common characteristic of machining of fibrous composites. This in turn can result in excessive surface roughness or poor tolerance of the machined part. It is reported in the literature that 60 percent of the composite parts rejected in the aerospace industry are as a result of delamination and hole quality problems (Isbilir & Ghassemieh, 2014).    1.3 Orthogonal Cutting in Metals In metal cutting, there exists a basic model which is referred to as the orthogonal cutting model. The setup of this model is as shown in Figure  1.1 along with the basic geometric features of the tool. The rake angle α is the angle between the vertical line and the rake face of the tool as indicated. This angle can be positive (as indicated in Figure  1.1) or negative. The relief angle γ is the angle between the horizontal plane and the bottom face of the tool as indicated. Rake and relief angle typically range between 5° to 20° and 5° to 7° respectively in orthogonal cutting experiments. The tool nose radius r and the cutting depth are also defined in the figure. 3   Figure ‎1.1_Basics of Orthogonal Cutting setup Chapter 1 A more detailed sketch of the orthogonal cutting setup specific to metal cutting is shown in Figure  1.2; in this model, a cutting tool cuts a constant depth of the material. A chip is formed through the shearing of the metal over the Primary (shear) Zone as indicated which is assumed to be a plane. A normal force and a frictional force act on the rake face of the tool over the Secondary Zone which sum up to the total force needed for shearing the constant depth of material and forming the chip. The total force acting on the tool however is also partially due to the forces induced as a result of the edge of the tool rubbing against the cut surface, referred to as the edge forces. The cutting forces which produce the chip are proportional to the depth and width of the material being cut while the edge forces are only proportional to the width of the material being cut as expected. By performing the orthogonal cutting experiment on a tool-workpiece material combination at different cutting depths and recording the horizontal forces (tangential or cutting forces) and the vertical forces (thrust or feed forces), a plot similar to the one shown in Figure  1.3 can be obtained. The slope of this plot yields the tangential and the feed (thrust) cutting constants denoted by     and     respectively; the intercept with the vertical axis on the plot yields the tangential and the feed edge constants referred to as     and     respectively. Cutting directionRake angle αRelief angle γCutting depthRake faceRelief face4   Figure ‎1.2_Orthogonal cutting of metal indicating the primary (shear) zone and the force decomposition     Figure ‎1.3_Experimental data indicating cutting and thrust force versus cutting depth (taken from Altintas, 2012) 5  Having found these constants for a given metal, force history plots in all other metal machining operations such as drilling, milling and turning are derived through a geometric and kinematic transformation of the orthogonal cutting model. By employing this approach, the force-time history in various machining operations can be obtained which can be used for the purpose of vibration and stability analysis during metal machining. Figure  1.4 summarizes the geometric transformation and the transition from orthogonal cutting to milling in metal machining. In short, cutting forces are obtained for tool-workpiece combination for different depth of cuts in order to obtain the cutting coefficients. By utilizing the cutting coefficients and discretizing the cutting edge of the tool in any machining operation into many small orthogonal cuts and superimposing all of them at the same time, the cutting force history can be obtained for a variety of machining operations. This can be used for purposes of obtaining the demands on the machining tool and machine and workpiece stability (Altintas, 2012).  Figure ‎1.4_ Transition from orthogonal cutting model to milling (adapted from Altintas, 2012) 1.4 Orthogonal Cutting in Composites For the purpose of investigating the machinability of composites and obtaining the machining forces as well as failure mechanisms involved in the machining of composites, researchers have applied the same orthogonal cutting approach used in metal to composites materials as well. In 6  the literature, there exists both experimental and numerical investigation of the orthogonal cutting of composites. Various parameters such as influence of fibre orientation and cutting depth on the cutting forces and the observed failure mechanisms are investigated in these studies. The experimental investigation by Zitoune et al. (2005) chooses the orthogonal cutting framework as a simpler approach to investigating the damage in fibre composites during drilling operations. Similarly, Bhatnagar et al. (1995) conducts multiple studies on the orthogonal cutting of composites as a way to gain further insight and into more complex operations such as drilling (Bhatnagar, Ramakrishnan, Naik, & Komanduri, 1995; Zitoune, Collombet, Lachaud, Piquet, & Pasquet, 2005).  1.5 Objective and Goals The aim of this study is to develop a numerical framework for evaluation of orthogonal cutting of composites. In this study, a thorough review of the literature is conducted on the fundamentals of composite machining and the significant parameters are identified and discussed. The goal of this research is to evaluate the use of finite element simulation of composite orthogonal cutting for obtaining cutting forces during orthogonal cutting of composites; orthogonal cutting models are developed in numerical platforms (ABAQUS and LS-Dyna) and the obtained results are compared with experimental data from other studies. This study is set to lay the foundation towards the ultimate objective which is to take the established framework in metal machining and apply it to machining operations in composites. This requires cutting theories and analysis techniques to be developed that suit cutting of composites.   1.6 Thesis Layout  The next two chapters of the thesis provide a thorough review of the existing work and material in the literature; Chapter 2 focuses mainly on the experimental studies while in Chapter 3, a review on the numerical studies is conducted. In Chapter 4 a description of the material models that were used in this numerical study are provided along with single element examples. The following two chapters (Chapter 5 and 6) explain the orthogonal cutting models in metals and composites in detail. In Chapter 7, a summary of the obtained numerical results (cutting forces) is provided from the numerical models developed in ABAQUS and LS-Dyna. Conclusions and some suggestions for future work based on the findings of this study are provided in Chapter 8.  7   Orthogonal Cutting of Composite: A Review of Experimental Studies Chapter 2In this chapter, a thorough review of the existing experimental work in the literature is provided; experimental studies in orthogonal cutting of composites are reviewed and parameters that most influence the cutting forces and chip formation, the two most fundamentals areas of study in composite cutting, are identified and discussed.  2.1 Introduction  Experimental investigations on the orthogonal cutting of composites go back as far as the 1980s. In most experiments, a laminate is created using unidirectional plies with the same fibre orientation; typical laminate thickness in most experiments is about 4 mm. A cutting tool with certain geometric specifications and material is used to cut a uniform depth of material on the edge of the laminate. This is usually conducted for fibres oriented in different directions with respect to the direction of the cutting tool movement; the most commonly investigated fibre orientations are 0°, 45°, 90° and 135° as shown in Figure  2.1. Most experiments use Polly Crystalline Diamond (PCD) tools as they provide excellent wear resistance. The geometric features of the tool include the rake angle and the relief angle. The rake angle can range from -20° to 20° depending on the experiment being performed. Despite some experiments being performed years ago, there is much consistency with the observations made in the more recent experiments. Researchers have reported many similarities in regards to chip formation mechanisms for various fibre orientations in the orthogonal cutting of composites. Many conclusions drawn from the experiments are similar which in turn helps with further developing the knowledge base in composite machining; however, at the same time there is also contradictions in the conclusions and findings, especially on the cutting forces obtained from the experiments, as well as the influence of tool geometry and experimental parameters on the cutting process (Ghidossi, El Mansori, & Pierron, 2006; Wang, Ramulu, & Arola, 1995,a; Wang & Zhang, 2003).  8   Figure ‎2.1_Fibre orientation in the orthogonal cutting of composites a) 0°, b) 45°, c) 90°, c) 135° (adapted from Wang et al. 1995,a)  2.2 Uni-Directional Composites  In this section, experimental studies on orthogonal cutting of uni-directional composites are reviewed. Fibre orientation tends to be the most important affecting the chip generation in composites; the subsequent subsurface damage as result of the cutting is also a function of the fibre orientation being cut. At the end of this chapter (Table  2-1), a list of some of the past experimental studies with some details regarding the specifications of the experiments is provided.     2.2.1 0° Fibre Orientation   The experimental work by Koplev et al. (1983) investigates the chip formation mechanism in orthogonal cutting of CFRP for two different fibre orientation of 0° and 90° as shown in Figure  2.2.   Figure ‎2.2_Experimental test setups, a) 0°, b) 90° (taken from Koplev et al. 1983) θ=0 θ=45 θ=90 θ=135a b c da b9  In the 0° fibre orientation where the fibres run parallel to the direction of the cut and are laid in the plane of the cut surface, the researchers report formation of cracks that run ahead of the currently produced chip along the fibre-matrix interface. Fracturing of fibres is observed to occur perpendicular to their longitudinal direction. This is very consistent with the observations made by Zitoune et al. (2005) The chip formation mechanism in 0° fibre orientation is as shown in Figure  2.3. As reported by Zitoune et al. (2005), with 0° fibre orientation, chips at the same thickness as the cutting depth split from the laminate without being deformed similar to a rigid body. It is believed that the cracks initiate through the micro-buckling of the fibres with compression reported as the dominant rupture mode. The tool creates an opening governed by mode I along with mode II with the formed chip analogous to a fixed-end beam under bending and compression. This opening produces a crack that propagates ahead of the formed chip according to multiple studies. Mkaddem et al. (2008) also reports propagation of a mode I crack and formation of a chip being deformed under bending action in the case of 0° orientation fibres; fracture is said to occur once the compressive strength of the fibre is reached (Koplev et al., 1983; Mkaddem, Demirci, & El Mansori, 2008; Venu Gopala Rao et al., 2007,b; Wang et al., 1995,a; Zitoune et al., 2005).  Figure ‎2.3_ Chip formation in 0° fibre orientation (adapted from Zitoune et al. 2005) This chip formation mechanism can in turn explain the splintering and fibre pull-out that occurs during composite machining, especially the drilling process. As the cutting edge of the drill bit 10  approaches a layer at 0°, it can cause a crack that runs ahead of the tool hence causing the splinters around the hole.  Wang et al. (1995,a) states that for 0° fibre orientation, opening mode I along with fracture governed by chip bending under cantilever load was observed; however, tools with zero or negative rake angles cause chip formation through micro buckling and fracture through mixed modes I (bending) and II (shearing along the fibre-matrix interface). In the case of 0° fibre, shearing occurs along the trim plane and the cut chip thickness is the same as the depth of the cut. The chips are also observed to have a considerable length to them; however, both the length of the formed chips and the height of the chips decrease as the fibre orientation changes from 0° to 90°. This will be further explained in the following sections. Another observation in this experimental study was the reduction in chip size with increasing fibre orientation and dust like chips beyond fibre orientation of 60°. For 15° fibre orientation, the matrix fracture patterns on the formed chips suggest interfacial shear fracture at the fibre matrix interface as the cutting mechanism involved in this orientation (Mkaddem e t al., 2008; Wang et al., 1995,a).    The length of the crack propagation depends on the depth of the material being cut. With increasing depth of cut, the crack propagates further. Once the formed chip, which is analogous to a beam under bending load, reaches a critical length, it fractures at the crack tip perpendicular to the fibre axes and the process repeats again in a cyclic manner (Zitoune et al., 2005).    In the case of 0° fibre orientation, Wang et al. (1995,a) states that as the tool geometry changes from a positive rake angle to a negative rake angle, the governing chip formation mechanism changes from a mode I opening to a mode II whereby micro buckling of the fibres becomes predominant. This is also illustrated in Figure  2.4 (Ghidossi et al., 2006; Wang et al., 1995,a). 11   Figure ‎2.4_Chip formation in 0° fibre orientation with a) positive and b) negative tool rake angles (adapted from Ghidossi, El Mansori, & Pierron 2006) The fibre orientation and the depth of cut are identified as the most significant parameters governing the chip formation and cutting forces in the cutting of composites. Wang & Zhang (2003) state that the influence from varying the rake angle of the tool is not as significant as that resulting from changing the depth of cut and fibre orientation. Tool geometry is known to have minimal influence on the orthogonal cutting of composites according to Venu Gopala Rao et al. (2007,b).  2.2.2 90° Fibre Orientation  As mentioned, chip formation in orthogonal cutting of composite is significantly influenced by the fibre orientation. In the case of the fibres running perpendicular to the cutting direction, the chip formation and cutting mechanisms are different from that of 0° fibres. In the cutting of 90° fibre orientation, cracks are formed parallel to fibres which run through the depth of the work piece; with increasing cutting depth, the cracks run further into the depth of the workpiece material hence causing extensive sub surface damage. This is also stated in experimental investigation by Wang et al. that subsurface damage becomes more severe with increasing cutting depth (Wang & Zhang, 2003; Zitoune et al., 2005).  In this setup (Figure  2.5), as the tool advances over the surface of the composite, fibres are compressed and slightly bent. The matrix cracks through the depth of the specimen under the applied compression resulting in significant sub-surface damage which extends below the cut surface; this compression induced rupture through the depth has been observed in experiments conducted by Venu Gopala Rao et al. (2007,b) The individual fibres, bending individually with θ=0 θ=0+ rake angle - rake angleα α• Mode I dominant chip formation • Fracture under beam bending and cantilever action • Mode II dominant chip formation • Fracture through micro buckling of fibres a) b)12  no coupling due to the matrix cracks, rupture through a mixed mode bending and shear and form non-continuous dust like chips. According to the study conducted by Wang et al. (1995,a), the fibre fracture mode in orientations beyond 75° consists of compression induced fracture normal to the fibre axes. Figure  2.5 illustrates the chip formation mechanism in the 90° fibres (Koplev et al., 1983; Venu Gopala Rao et al., 2007,b; Zitoune et al., 2005).  The surface roughness tends to be higher when cutting perpendicular to the fibres with an uneven surface profile. It is reported by multiple studies that in cutting of 90° and higher fibre orientations, much of the fibres are bent without being cut and as the tool advances; the uncut fibres bounce back and remain on the surface yielding an uneven surface p1rofile. Pwu & Hocheng (1998) however, states that the fibres are cut from a point below the cut surface; this is somewhat contrary to the findings by Venu Gopala Rao et al. (2007,b) which shows that the peak stress in the fibres and subsequently the point of fracture are about 35 μm above the cut surface. However, it must be noted that other factors such as shear strength of the matrix, cutting edge radius and other cutting parameters can have significant influence on the surface profile (Bhatnagar et al., 1995; Iliescu, Gehin, Iordanoff, Girot, & Gutierrez, 2010; Pwu & Hocheng, 1998; Venu Gopala Rao et al., 2007,b).  Figure ‎2.5_ Chip formation in 90° orientation (adapted from Zitoune et al. 2005) 2.2.3 45° Fibre Orientation  In the case of fibres aligned at 45°, the tool shears the fibre and the matrix perpendicular to the fibre axis through shear loading; the fracturing of the fibres through this process yields the 13  formation of the chip. The chip releases along the fibre-matrix interface. Zitoune et al. (2005) noted dependency of the formed chip on the depth of cut for the 45° fibre orientation; for depths of cut lower than 0.2 mm, a continuous ribbon-like chip was formed as opposed to the dust-like chips observed in most experiments and fibre orientations. Higher depth of cuts yielded the more commonly reported dust-like chips (Ghidossi et al., 2006; Wang et al., 1995,a; Zitoune et al., 2005). This chip formation process explained here is generally attributed to fibre orientations ranging from 15° to 75°. This includes fracturing of fibres through compression induced shear and release of chip along the fibre-matrix interface as shown in Figure  2.6 (Ghidossi et al., 2006; Wang et al., 1995,a).    Figure ‎2.6_Chip formation mechanism in 45° fibre orientation (adapted from Wang et al. 1995,a) 2.2.4 135° and Greater Fibre Orientations  In the case where the fibres are orientated such that the tool advances opposite to the flow of fibres, the general fracturing mode of fibre is through compression induced rupture and the subsequent chip release is through fibre-matrix interface debonding. In such high fibre orientations, Wang et al. (1995,a) reports extensive out of plane displacement and material fracture ahead of the cutting tool (Wang et al., 1995,a). Wang & Zhang (2003) note that in the orthogonal cutting of fibre orientation greater than 90°, many fibres are bent, but not cut; as the cutting tool approaches, the uncut fibres bounce back elastically. Bhatnagar et al. (1995) states that in fibres oriented above 90°, the fibres tend to bend to the underside of the tool; the fibres hence leave an uneven cut plane full of uncut fibres. Delamination, and formation and propagation of deep cracks into the depth of the specimen are listed as the characteristics of cutting greater than 90° fibre orientations. It is interesting to note θ=45 14  the similarities between the observations reported by different researchers (Bhatnagar et al., 1995; Wang & Zhang, 2003). Wang et al. (1995,a) studies surface roughness of machined unidirectional CFRPs and summarizes the fibre fracture process in the following manner:   For fibre orientations less than 90°, as the cutting tool advances, the fibres being pushed are well supported by the material behind; hence the component of force parallel to the fibres induces tensile stress in the fibres and causes fracturing of the fibres   For fibre orientations greater than 90°, there is less support from the backing material as the fibre undergoes bending due to advancement of the tool, which in turn causes a more severe fibre-matrix debonding Unfortunately, amongst the limited literature available on the orthogonal cutting of composites, very few study fibre orientations greater than 90°.  2.3 Chip Generation  The chips are of small size and dust-like according to Wang et al. (1995,a) in the machining of greater than 60° fibre orientation; for fibre oriented less than 60°, the chips have a certain length to them. Bhatnagar et al. (1995) also reports formation of blocky chips in less than 90° fibre orientation. Cracks on the cut surface and out of plane displacements ahead of the tool are commonly reported in the machining of composites, especially in fibre orientations greater than 90°. The cutting of fibre with orientations great than 90°, such as 135° shows delamination and macro fractures ahead of the cutting tool (Bhatnagar et al., 1995; Wang et al., 1995,a). In general, when the fibres are oriented along the direction of the tool movement (less than 90° fibre orientations), the fibres undergo a clean cut along their transverse axes as stated by multiple studies; however, in fibre orientations where the fibres are aligned opposite to the direction of tool movement, severe fibre matrix deboning occurs and the process yields an uneven cut surface (Nayak, Bhatnagar, & Mahajan, 2005,a).  In the machining and orthogonal cutting of composites, rupture mode of the fibres depends of the fibre orientation; Pwu & Hocheng et al. (1998) however suggests that rupture occurs as a result of bending induced tensile stresses; it is stated that since the individual fibre diameters are less 15  than the cutting edge radius, 7-10 μm versus 10-20 μm cutting edge radius, shearing of fibres cannot occur. Venu Gopala Rao et al. (2007,b) indicates through numerical analysis that fibre failure occurs as a results of local bending and tensile failure. Bhatnagar et al. (1995) reports fibre tensile failure followed by shearing of the matrix along the fibre matrix interface as the cutting mechanism in fibres oriented between 0° and 90°. Bhatnagar et al. (1995) and Wang et al. (1995,a) both state the importance of the in-plane shear strength of the uni-directional composites in the material removal during orthogonal cutting. The existing work in the literature however, is mostly limited to qualitative analysis and observations made during experiments and very few data exist where the focus is on quantifying the mechanisms and cutting characteristics (Bhatnagar, Ramakrishnan, Naik, & Komanduri, 1995; Pwu & Hocheng, 1998; Venu Gopala Rao, Mahajan, & Bhatnagar, 2007,b; Wang, Ramulu, & Arola, 1995,a). 2.4 Bouncing Back Phenomena In orthogonal cutting of composites, the nominal depth of cut tends to vary from the real depth of cut as a result of uncut fibres bouncing back elastically and remaining on the cutting plane; the variation is dependent on the depth of cut and the fibre orientation. Wang & Zhang (2003) compare the real depth of cut with the nominal depth of cut for multiple fibre orientations and various cutting depths. In the experimental study by Iliescu et al. (2010), it is stated that after shearing of the 90° fibres, some fibres recover elastically and rub against the underside (relief face as defined in Figure  1.1) of the tool. It is found that for small depths of cut, below 100 μm (0.10 mm), the bounce back can be up to 50 μm; for higher depths of cut, the bounce back length remains almost unchanged hence indicating a threshold where the bounce back reaches its limit. The researchers also find that the measure of bounce back is related to the cutting edge radius as defined in Figure  1.1; for fibre oriented less than 90°, the bounce back is equal or slightly greater than the edge radius of the tool whereas in higher fibre orientations, the bounce back can be up to twice the cutting edge radius (Iliescu et al., 2010; Wang & Zhang, 2003).  Venu Gopala Rao et al. (2007,b) shows through numerical simulation that in the trimming of 90° fibre orientation with a tool with a cutting edge of 50 μm radius, the fibre reaches its tensile strength at a point about 40 μm above the trim plane hence fracturing the fibre from a point above the trim plane and leaving behind uncut material. Bhatnagar et al. (1995) reports observing the same phenomenon when cutting greater than 90° fibre orientation; it is mentioned that fibres 16  are bent to the underside of the tool edge and are not cut in the process. Although Pwu & Hocheng (1998) do not specifically mention bounce back, they state that in the case where the cutting edge radius is less than the diameter of the fibres, they cannot be sheared in the cutting process and instead, they undergo bending; hence, making reference to the cutting edge radius and the chip formation process (Bhatnagar et al., 1995; Pwu & Hocheng, 1998; Venu Gopala Rao et al., 2007,b). 2.5 Forces in The Orthogonal Cutting of Composites  One of the important aspects of conducting orthogonal cutting experiments on composites apart from the chip formation mechanism is to obtain the cutting forces. Experimental observations have shown that cutting forces in the orthogonal cutting of composites depend on the fibre orientation and the tool geometry with the former having a much more significant impact. The forces involved in orthogonal cutting are typically decomposed into two components; the horizontal component which is in the direction of the cut and along the cutting plane and is referred to as the cutting forces and another component which is perpendicular to the cutting direction referred to as the thrust forces. This is also graphically illustrated in Figure  1.2. 2.5.1 Cutting Forces Venu Gopala Rao et al. (2007,a) investigate the influence of fibre orientation, rake angle and cutting depth on the induced cutting forces during the orthogonal cutting of both unidirectional CFRP and unidirectional GFRP. It is reported that the cutting forces are most dependent on the fibre orientation and on the depth of the material being cut and less on the rake angle of the tool. This is very much in line with the findings of Wang et al. (1995,a) as well as other experimental studies stating that the cutting forces, as well as the chip formation and the roughness of the cut surface in unidirectional laminates are dependent on the fibre orientation with moderate influence from the tool geometry (Venu Gopala Rao, Mahajan, & Bhatnagar, 2007,a; Wang et al., 1995,a).  As the fibre orientation varies from 15° to 90°, the range investigated by Venu Gopala Rao et al. (2007,b), the cutting forces (horizontal forces) increase according to the plot shown in Figure  2.7. Looking at the variation of the cutting force versus the fibre orientation for the two different materials, namely the UD-CFRP and the UD-GFRP, the plots are almost parallel for all cutting depths. This suggest that the cutting forces increase steadily with increasing fibre orientation, 17  and that at least for positive angles of cut, the cutting forces versus the fibre orientation follows the same trend regardless of the type of material or the depth being cut. This observation indicates that the evolution of the cutting forces with the fibre geometry is independent of the material being cut. This in turn can be an indication that the cutting and material removal mechanisms in fibre reinforced composites are the same regardless of the material. However, more evidence and experimental data is required in order to arrive at a concrete conclusion in this regard. Also, it must be noted that based on the experimental observations, the chip formation mechanism depends on the fibre orientation as previously explained.  Another observation made by Venu Gopala Rao et al. (2007,b) is that the failure of fibres is governed mainly by a combination of crushing and bending. With the fibre orientation varying from 90° to 15°, the degree of bending failure increases. The degree of sub-surface damage that extends below the cutting plane is reported to be the highest for the 90° fibre orientation; this observation is consistent with other studies of similar type (Venu Gopala Rao et al., 2007,b).  Figure ‎2.7_Cutting forces versus fibre orientation in CFRP and GFRP orthogonal cutting for multiple depths of cuts (taken from Venu Gopala Rao et al. 2007,b) 18  In the CFRP orthogonal cutting experiments conducted by Bhatnagar et al (1995).; the cutting force data for the two different rake angles used in the experiments are gathered and plotted Figure  2.8. As shown, the cutting force show a very gradual raise through up to 90° with a sudden increase of forces at 90° followed by reduction of cutting force for higher fibre orientations for both rake angles tested.   Figure ‎2.8_Cutting force versus fibre orientation for experiments conducted by Bhatnagar et al. (1995) CFRP material with cutting depth of 0.25 mm and cutting width of 2.2 mm Wang et al. (1995,a) investigates the chip formation mechanisms as well as measuring the cutting and thrust forces on unidirectional CFRP with different fibre orientation; the cutting and thrust force plots obtained from the experiments performed are illustrated in Figure  2.9. The average cutting forces (defined as principal forces in the plots) increase gradually as the fibre orientation goes up from 0° to 75°, but increase largely when cutting fibre at 90° orientation and decrease again for fibre orientations greater than 90°. This is also the case In the CFRP orthogonal cutting experiments conducted by Bhatnagar et al (1995).  The cutting (principal) force fluctuations vary greatly for each fibre orientation both in magnitude as well as the frequency of the fluctuations. It is reported that with increasing fibre orientation, the cutting force fluctuations decrease until a fibre orientation of 90°, at which point 0204060801001201401600 30 60 90 120 150 180Cutting Force (N/mm)Fibre orientation (degrees)18° rake angle12° rake angle19  the nature of the force fluctuations change to a higher frequency. According to Wang et al. (1995,a) and Nayak et al. (2005,a), the high frequency variation of the cutting forces is due to the repeated fibre/matrix microstructure of the composite material; the fluctuations are due to the fibres being much stiffer than the matrix. (Nayak, Bhatnagar, & Mahajan, 2005,a; Venu Gopala Rao, Mahajan, & Bhatnagar, 2007,b; Wang, Ramulu, & Arola, 1995,a).   Figure ‎2.9_Cutting force plots for multiple fibre orientations in orthogonal cutting of CFRP; cutting depth of 0.25 mm, cutting width of 4 mm, tool rake angle 10° (taken from Wang et al. 1995,a)  The cutting force data from studies by Koplev et al. (1983) and Zitonue et al. (2005) for 0° fibre orientation are gathered and plotted as in Figure  2.10. The gathered data are obtained from orthogonal cutting of CFRP at 0° for multiple depths of cut; despite the slight differences the 20  cutting speed, tool geometry and material properties, the forces are within a narrow range and show a similar trend with increasing depth of cut. It must be noted that the cutting forces increase proportionally with increasing cutting depth for the range of cutting depths tested. This is expected and indicates the presence of the same cutting mechanism in all depths of cut.   Figure ‎2.10_Cutting force vs. Cutting depth for 0° fibre orientation (adapted from Koplev et al. 1983 and Zitonue et al. 2005) As stated, in fibre composites, cutting forces are highly dependent on the fibre orientation relative to the cutting direction; hence, as opposed to isotropic metals where the cutting force is independent of cutting direction, in fibre composites, cutting forces for each fibre orientation must be evaluated in order to be able to predict them in more complex machining operations as the cutting edges of the tool come in contact with multiple fibre orientations in each revolution. Unfortunately, very limited data is available on the cutting forces and the cutting force history plots from experimental measurements is rarely provided in the literature.  In most literature available, the cutting forces show a steady increase from 0° to just before 90° fibre orientation followed by a sudden jump in the cutting forces at 90°. This can be seen in the results obtain by Bhatnagar et al. (1995) and Wang et al (1995,a) (Figure  2.9); no specific 010203040506070800 0.05 0.1 0.15 0.2 0.25Force (N/mm)Cutting depth (mm)(Koplev et al., 1983) 0° rake(Zitoune et al., 2005) 6° rake(Koplev et al., 1983) 10° rake21  comments have been made on this particular observation. However, this is not the case in the results obtained from experiments conducted by Venu Gopala Rao et al. (2007,b); as indicated in Figure  2.7 which shows a steady increase in the cutting forces up to 90° in the orthogonal cutting of both CFRP and GFRP. This is also the case in the experiments conducted by Nayak et al. (2005,a) in the orthogonal cutting of GFRP which reports a very small increase of cutting forces from 0° to 90° fibre orientation.  Nayak et al. (2005,a) reports a slight increase in the cutting forces at different depths of cut as shown in Figure  2.11 which was obtained from this study; the moderate increase in the cutting forces is attributed to the increasing volume of matrix that has to be removed and that the magnitude of the force needed to break the fibres is independent of the depth of the material being cut as the fibres fail upon reaching their failure strength at the cutting plane (Nayak, Bhatnagar, & Mahajan, 2005,a). Wang et al. (1995,a) on the other hand notices a linear increase in the cutting as well as the thrust forces with increasing depth of cut. Depth of cut and fibre orientation are known to be the most significant parameters influencing the machining forces (Wang et al., 1995,a).    Figure ‎2.11_Cutting forces (N/mm) in orthogonal cutting of GFRP for multiple fibre orientations and cutting depths of 0.1, 0.2 and 0.3 mm (taken from Nayak, Bhatnagar, & Mahajan 2005,b) 2.5.2 Thrust Forces  Thrust force is the component of force that acts perpendicular to the cutting direction. In most experiments, only the cutting forces are measured and reported. In the numerical studies, the 22  trends predicted for thrust forces are usually off from those obtained from the experiments. In a typical drilling operation, the cutting forces determine the torque about the longitudinal axis of the tool while the thrust forces at the cutting edge determine the total thrust force acting on the tool and the workpiece; it is of great important to be able to predict thrust forces especially in composites as they induce delamination which is one of the main concerns during machining of composites.  In the study by Koplev et al. (1983), the cutting and thrust forces are obtained only for 0° fibre orientation. The study is conducted for tools with different geometries and forces vary greatly depending on the changing parameters. However, generally the thrust forces are substantially higher than the cutting forces for 0° fibre orientation. Similar observation is reported in the study by Wang et al. (1995,a); for all fibre orientations up to 75° and for tools with positive rake angles, the average thrust forces during the cut was higher than the cutting forces which are in contrary to the expectations of the authors. Further details regarding the dependency of these forces on the changing parameters are provided in Section  2.5.3.   The study by Venu Gopala Rao et al. (2007,b) shows a steadily decreasing thrust force with increasing fibre orientation from 0° to 90° as shown in Figure  2.12. The thrust forces are lower than the corresponding cutting forces for most fibre orientations in this study; however, this contradicts what has been reported by Wang et al (1995,a).  23   Figure ‎2.12_ Machining forces versus fibre orientation in CFRP and GFRP orthogonal cutting for cutting depth of 0.1 mm (adapted from Venu Gopala Rao, Mahajan, & Bhatnagar 2007,b) Nayak et al. (2005,a) conducts an extensive study on the orthogonal cutting of GFRP composites along with a thorough parametric study on the effect of tool geometry on the cutting forces. The trend for the cutting forces as well as the thrust forces obtained from this study is presented in the plot in Figure  2.13. Contrary to what was shown by Venu Gopala Rao et al. (2007,b) in Figure  2.12, the thrust forces in this experimental study show an increase up to 45°, at which point they start decreasing again. The cutting force, however, shows a similar trend as previously stated.  010203040500 15 30 45 60 75 90Force per unit width (N/mm)Fibre orientation (degree)CFRP Cutting ForceCFRP Thrust ForceGFRP Cutting ForceGFRP Thrust Force24   Figure ‎2.13_Machining force versus fibre orientation in GFRP orthogonal cutting for cutting depth of 0.2 mm adapted from Nayak et al. (2005,a) and for CFRP orthogonal cutting for cutting depth of 0.25 mm (adapted from Wang et al. 1995,a) 2.5.3 Influence of Tool Geometry  In this section, the influence of the tool geometry, namely the rake angle and the relief angle of the tool as defined in Figure  1.1 on the cutting forces will be discussed.  2.5.3.1 Rake Angle  Koplev et al. (1983) conducts a parametric study to investigate the influence of the rake and relief angle of the tool (as defined in Figure  1.1) on the cutting and thrust forces in 0° fibre orientation; rake angles ranging from 0° to 10° and relief angles ranging from 5° to 15° are used in this study. It is stated that generally there is a relation between the rake angle and the cutting forces and between the relief angle and the thrust forces during orthogonal cutting. The increase in the rake angle causes a slight decrease in the cutting force as the chips flow more easily on a shallower rake face. Wang et al. (1995,a) also reports reduction in the cutting forces in 0° fibre orientation with increasing rake angle as this causes a shift in the chip formation from shear and micro-buckling mode II to peeling mode I (Koplev et al., 1983; Wang et al., 1995,a). 0204060801001201401601802000 15 30 45 60 75 90Force per unit width (N/mm)Fibre orietnation (degree)Cutting forces, Nayak et al. (2005,a)Thrust forces, Nayak et al. (2005,a)Cutting forces, Wang et al.(1995,a)Thrust forces, Wang et al.(1995,a)25  Nayak et al. (2005,a) and Wang et al. (1995,a) explore the influence of the geometric parameters of the tool much more thoroughly in the orthogonal cutting of GFRP and CFRP. The general observation is that increasing rake angle results in decreasing cutting forces. Rake angles tested in the experiment by Nayak et al. (2005,a) cover a much larger range compared to other experiments of similar nature as they range from 0° to 40°. The cutting forces decrease with increasing rake angle for all fibre orientations with the minimum cutting force recorded at a rake angle of 30°. The data obtained for the rake angles tested is shown in Figure  2.14; as it can be seen, the relative change in the forces varies from one fibre orientation to another with 90° fibre orientation showing greater sensitivity to changes in rake angle compared to 30° fibre orientation. This is in contrary to the results obtained by Wang et al. (1995,a) which indicate the reduction in the effect of rake angle with increasing fibre orientation; that is to say the cutting forces do not change as much with variation in the rake angle of the tool for higher fibre orientations such as 75° and 90° fibre orientations (Nayak et al., 2005,a; Wang et al., 1995,a). Wang & Zhang (2003) report small reduction in the cutting forces with increasing rake angle for a cutting depth of 0.050 mm; however, the variation in the cutting forces due to rake angle variation is very moderate and the greatest influence on the machining forces come from fibre orientation and cutting depth. Wang et al. (1995,a) also state that the influence of tool geometry and operating conditions are much less than that of the fibre orientation. Other literature also report minimal influence from the tool geometry on the machining forces in the orthogonal cutting of fibrous composites (Nayak et al. , 2005,a; Wang et al., 1995,a; Wang & Zhang, 2003).   26   Figure ‎2.14_Cutting forces versus the rake angle of the tool for multiple fibre orientations at 0.2 mm depth of cut (adapted from Nayak et al. 2005,a) 2.5.3.2 Relief Angle  Although changes in the rake angle cause the thrust forces to vary for different depths of cut, no trend is obvious and no comments are made by the author in the experiments by Koplev et al. (1983) Increasing the relief angle however, results in a drastic drop in the thrust forces; it is stated that this is due to the reduced contact area between the tool underside and the workpiece as the relief angle increases. The cutting forces however remain constant and nearly unchanged with changes in relief angle. This finding is very consistent with what has been reported by Wang et al. (1995,a) regarding no change in the cutting forces and reduction of the thrust forces as a result of increasing relief angle (Koplev et al., 1983; Wang et al., 1995,a).  2.5.3.3 Cutting Edge Radius Multiple studies exist where the cutting forces are obtained for different cutting edge radii; the results indicate that sharper cutting edges with smaller radii yield reduced cutting forces. Nayak et al. (2005,a) conduct experiments with tools with edge radii of 20, 50 and 80 μm for two 2030405060700 10 20 30 40Cutting force per unit width (N/mm)Rake angle of tool (degree)90 Fibre orientation60 Fibre orientation45 Fibre orientation30 Fibre orientation27  cutting depths of 0.1 and 0.2 mm. It is observed that both cutting and thrust forces are slightly higher for an edge radius of 80 μm and cutting depth of 0.1 mm. In cutting depth of 0.2 mm, no change in any of the cutting forces for any fibre orientation is present for the different edge radii. Venu Gopala Rao et al. (2008) also state marginal reduction in the cutting forces due to increase in rake angle; the reduction in the forces is attributed to bending and wrapping of the fibres around the cutting edge of the tool in order to remain in touch with the rake face of the tool (Nayak et al., 2005,a; Venu Gopala Rao, Mahajan, & Bhatnagar, 2008).  2.6 Multidirectional Composites  The experimental observations and findings discussed up to this point corresponded to orthogonal cutting of unidirectional laminates. In industry however, composite laminates tend to be multidirectional and hence it is of great importance to investigate the machinability of such laminates as well. Wang et al. (1995,b) performed orthogonal cutting on 4 mm thick IM-6 graphite epoxy laminates with layup                               .  This study was conducted alongside the experimental investigation on uni-directional laminates and comparisons were made on the chip formation mechanism in each ply of the multidirectional laminate and the uni-directional laminate. 2.6.1 Chip Formation  The failure in the 0° plies in the multidirectional laminate was observed to be similar to that of 0° unidirectional laminate where fracture is governed by mode I bending and mode II shearing. Chips in the 45° orientated plies were formed through compression induced shear at the contact zone and fibres fractured perpendicular to their axis which is identical to that observed in the cutting of 45° unidirectional composite. Although the chip formation in the 90° and -45° (or 135°) fibre orientation were observed to be similar to that reported in uni-directional cutting, the extent of subsurface and out-of-plane damage present in each ply was substantially reduced due to the support provided by the adjacent plies. In the 90° ply, the fibre-matrix debonding was limited to the trim surface and did not extent into the depth of the material as it was observed in trimming of uni-directional 90° composite. Although the fibre matrix interface debonding and the subsequent sub-surface damage in the -45° ply was more extensive, the extent of this damage was limited in the multidirectional laminate (Wang, Ramulu, & Arola, 1995,b).  28  2.6.2 Cutting Forces The cutting forces obtained from the orthogonal cutting of the multidirectional laminate by Wang et al. (1995,b) indicates force fluctuations similar to that observed in the trimming of 90° fibre orientation as shown in Figure  2.9. It is stated that the cutting forces in the multidirectional laminate is nearly equal to summing the forces per unit width of each ply obtained from trimming the uni-directional plies separately. Also, due to the similar nature in the force fluctuation with that of the 90° fibre orientation, it is concluded that the removal mechanism is governed by this fibre orientation (Wang et al., 1995,b).  2.7 Summary The field of composite machining is clearly not very well developed despite the attention it has been receiving from 1980’s onwards from the academic side as well as interest from the industry. Based on the available literature, there has been much focus on the orthogonal cutting of unidirectional laminates and much experimental observations exists on the cutting and chip formation mechanisms as well as the type of damage that occurs in cutting of each fibre orientation. A summary of the available experimental studies in the literature is provided in Table  2-1. The shortcoming however is in relating the orthogonal cutting to machining operations of interest. There is no study in the literature where the results from an orthogonal cutting operation are linked to a machining process such as drilling or milling. As explained, in metal machining, the orthogonal cutting process is translated into milling and drilling operations by applying a geometric transformation of the orthogonal cutting. In theory, any cutting operation consists of superimposing many small orthogonal cutting operations occurring at a given location and time. Despite the literature taking the same framework as in metals industry, no attempts have been made to investigate the relation between orthogonal cutting and another machining operation.     Also, another step that is missing from the literature is the lack of experimental data on orthogonal cutting of multidirectional laminates or unidirectional laminates with extra adjacent layers in order to obtain damage propagation data that better represent the cutting condition during operations of interest such as drilling and milling. As reported by Wang et al. (1995,b), the damage propagation on certain lamina orientations varied depending on their application in unidirectional or multidirectional laminates. Therefore, it would be of great benefit to further 29  explore this field by comparing the cutting force and damage propagation results from unidirectional and multidirectional laminates and seek for any correlation or differences between the two.  Based on what is discussed and thought to be missing from the literature, it would be very insightful to perform a set of orthogonal cutting experiments and obtaining the required cutting forces for multiple fibre orientations and perform a drilling experiment on a laminate of the same materials. Using the orthogonal cutting results and by employing the same geometric transformations used in metal machining, the theoretical torque history for the drilling operation can be obtained and compared with the experimental results. Comparing the two torque history plots, one derived from a combination of orthogonal cutting and cutting theories and another obtained entirely through experiments will give great insight into the cutting process and mechanism. Doing so will allow one to confirm that the cutting mechanisms during the orthogonal cutting process are really representative of the mechanisms that take place during other machining operations.             30  Table ‎2-1_ List of experiments on orthogonal cutting of composites Author Material Tool Experiment Specifications  (Koplev, Lystrup, & Vorm, 1983) UD CFRP  Epoxy resin  High Speed Steel  α = 0, 5, 10, γ = 5, 10, 15 Speed = 26.4, 48 m/min Cut. Dep. = 0.05, 0.1, 0.2 mm (Wang, Ramulu, & Arola, 1995,a)  CFRP Unidirectional  IM-6/3501-6 PCD tool insert  α =0, 5, 10, γ = 7, 17  Speed = 4, 9, 14 m/min Cut. Dep. = 0.127, 0.254, 0.381 mm  (Bhatnagar, Ramakrishnan, Naik, & Komanduri, 1995) UD CFRP  Epoxy resin  Brazed carbide tool (Grade K-20) α = 12, 18 Speed = 1.18 m/min Cut. Dep. = 0.25 mm  (Pwu & Hocheng, 1998) UD CFRP Not Specified  Speed = 1, 3, 6 m/min Cut. Dep. = 0.05, 0.1, 0.2 mm  (Wang & Zhang, 2003)  UD CFRP  F593 and MTM56 prepregs     Tungsten carbide tool α = 0, 20, 40, γ = 7  Speed = 1 m/min Cut. Dep. = 0.025-0.250 mm (inc. 0.025mm) (Zitoune, Collombet, Lachaud, Piquet, & Pasquet, 2005) UD CFRP  T2H/EH25 Tungsten carbide tool insert  NEF 66 366  α =7, γ = 6  Speed = 0.5 m/min Cut. Dep. = 0.05, 0.25 mm (Nayak, Bhatnagar, & Mahajan, 2005,a) UD GFRP Epoxy resin HSS and Carbide (Grade K-10) tool  α = 0, 5, 10, 20, 30, 40 γ = 6 Speed = 0.5 m/min Cut. Dep = 0.1, 0.2, 0.3 mm (Venu Gopala Rao, Mahajan, & Bhatnagar, 2007,b) UD CFRP and GFRP  LY 556 epoxy    Solid tungsten carbide (Grade K-10) α = 5, 10, 15, γ = 6 Speed =0.5 m/min  Cut. Dep. = 0.1, 0.15, 0.2 mm  (Dandekar & Shin, 2008) UD CFRP by Hexcel Inc.   Solid Carbide tool  α = 5, γ = 6 Speed = 60 m/min (1 m/s) Cut. Dep. = 0.1 mm (Iliescu, Gehin, Iordanoff, Girot, & Gutierrez, 2010) UD CFRP Carbide tool α = 0 Speed = 6 m/min  Cut. Dep. = 0.2 mm 31   Orthogonal Cutting of Composites: A Review of Numerical Studies Chapter 3With increasing applications of FRP composites in different industries, especially the aerospace industry, there is much interest to develop an understanding of the field of machining. In doing so, experiments provide much insight into the chip formation process and the cutting mechanisms involved in the machining of composites; however, experiments are expensive to run, require additional equipment for purposes of dust evacuation and pose health risks to those performing the experiment. Hence numerical approaches are of particular interest to study the machining of composites. The numerical tools are used to obtain cutting forces and thrust forces in machining processes as well as chip formation mechanisms involved in composite cutting. As mentioned previously, composite machining is a relatively underdeveloped field; this is especially true in the field of numerical modeling. In this chapter, some of the numerical studies on the orthogonal cutting of composites will be presented and reviewed. 3.1 Introduction  A number of numerical studies are available in the literature on the orthogonal cutting of composites with each numerical representation employing a particular material and damage model, element type and tool geometry which sets them apart. There are two general approaches for numerical modeling of composites orthogonal cutting. Macro-Mechanical and Micro-Mechanical approaches. In the former, Macro-Mechanic, the composite workpiece is represented by an Equivalent Homogeneous Material (EHM) with orthotropic material properties. This approach is commonly employed by many of the researchers as it tends to be simple to implement and yet yields reasonably accurate predictions according to the literature. The latter of the two approaches, Micro-Mechanical approach, requires modeling of the fibres and the matrix phase individually and implementing an interaction between the two phases of the composites. This approach comes at a cost of increased computational time and effort; however, it has been shown that more accurate results can be obtained, especially in predicting the thrust forces during orthogonal cutting. ABAQUS Explicit is amongst the most popular numerical platforms used to investigate the orthogonal cutting of composites. Details specific to each modeling approach are explained in the following sections. Table  3-1 provides a list of numerical studies on the orthogonal cutting of composites.   32  3.2 Modeling Approach   Few studies exist in the literature which conduct numerical studies of composite orthogonal cutting. In most of these studies, the numerical representations of composite orthogonal cutting is in two dimensions and employs plane stress elements as opposed to plane strain which is used in modeling of metal machining. Arola et al. (2002) and Nayak et al. (2005,b) both justify the use of plane stress in composite based on the experimental observations where out-of-plane displacements during the cutting process were noted. As mentioned earlier, in general two main numerical approaches exits; micro-mechanical approach in which the fibre, matrix and the interface between the two is considered in the model, and the macro-mechanical approach in which the composite is represented by an Equivalent Homogeneous Material (EHM). Further details are provided in the following sections (Arola, Sulta, & Ramulu, 2002; Dandekar & Shin, 2008; Nayak et al., 2005,b). 3.2.1 Macro-Mechanical Approach In the macro-mechanical approach, the inhomogeneous nature of the material is ignored and the fibre/matrix composite is represented by an equivalent orthotropic homogeneous material. There are studies (Table  3-1) which indicate a reasonable prediction of cutting forces can be obtained using this method; however, there are a few shortcomings associated with the homogeneous representation of the composites. Although this approach yields reasonable predictions of the cutting forces, the thrust forces tend to vary significantly in both the value and the trend predicted for each fibre orientation (Soldani, Santiuste, Munoz-Sanchez, & Miguelez, 2011). Also, it has been suggested by Venu Gopala Rao et al. (2007,b) that this method fails to give insight into the cutting mechanisms and fails to explain the cutting process at the micro-level. On the other hand, this model is much easier to implement and due to its simplicity compared to the micro-mechanical models, it is computationally less intensive. It must be noted that the material model used to represent the composite as a homogeneous material can have different levels of complexity for capturing damage progression (Venu Gopala Rao et al., 2007,b). 3.2.2 Micro-Mechanical Approach  In the micro-mechanical approach, the composite is represented thorough modeling the fibre phase and the matrix phase with the boundary between these two phases represented by an interface governed by a separation criterion. It is very common to use cohesive elements with a 33  traction-separation law where a stress-displacement law is assigned with a certain fracture energy. The cohesive zone allows for complete separation upon reaching a saturation value for a displacement. Some of the more recent studies use a micro-macro combined representation of the workpiece material in which the vicinity around the tool-chip contact is modeled using the micro-mechanical approach and further away from this region a macro-mechanical representation of the material is used to model the entire workpiece.  Venu Gopala Rao et al. (2007,b) and Nayak et al. (2005,b) both use this approach in numerical modeling of composite orthogonal cutting. Both studies obtained good agreement between the numerical and the experimental studies for both the cutting forces and the thrust forces which the macro-mechanical model fails to predict accurately. They also explain some of the experimental observations such as bounce back and chip removal mechanisms using the stress distributions found in the micro-mechanical models at the micro-scale; according to Dandekar et al. (2008), micro-mechanical models allow for evaluation of local defects such as fibre-matrix debonding. More specific details about the models and the outcomes is provided in the following sections (Dandekar & Shin, 2008; Nayak et al., 2005,b; Venu Gopala Rao et al., 2007,b).    3.2.3 Input Parameters    Despite the differences in the material models implemented in each study and the preferred modeling technique, and geometry of the tool and workpiece material, there are a few input parameters and assumptions that are consistently present in most studies:    Cutting tool considered rigid   Coulomb friction assumed between the tool and the workpiece with friction coefficient between 0.3-0.6  Heat generated during the cutting process is neglected as the cutting speeds at which experiments are conducted are low and temperatures do not become high enough to cause changes in the mode of fracture or matrix viscosity (Abena, Leung Soo, & Essa, 2015; Arola et al., 2002)  Base of the workpiece is restrained in the cutting and vertical directions and the sides are restrained in the cutting direction only as illustrated in Figure  3.1.     34   Figure ‎3.1_Workpiece boundary conditions for purpose of numerical analysis    Cutting depths usually in the range of 0.1-0.2 mm  Tool rake and relief angles, α and γ, tend to be between 5°-7° in the numerical simulations 3.3 Macro-Mechanical Models   In the following sub-sections, the damage models implemented in macro-mechanical representations of orthogonal cutting is explained. Most of the existing models in orthogonal cutting is developed in a two-dimensional space; very few studies exists that models composite orthogonal cutting in a three-dimensional space. This section covers studies in both two- and three- dimensions.  3.3.1 Two-Dimensional   In some studies (Arola & Ramulu, 1997; Nayak, et al., 2005,b) which take the macro-mechanics approach, failure and damage is limited to a predefined zone along the plane of the cut. In the numerical study by Nayak et al (2005,b), the model developed using the macro-mechanics approach, failure is assigned to the nodes located on the trim plane along the cutting path region and to the workpiece material itself. Therefore a pre-defined path for the separation of the chip from the trim plane is defined by assigning a separation criterion to the nodes located along the Fibre orientation θCutting directionRake angle αrelief angle γXY35  trim plane. The separation criterion implemented by Nayak et al. (2005,b) is purely strength-based and follows Equation ( 3-1).    √(     )  (    )   (‎3-1) where    and   are the normal and shear stresses along the predefined trim plane and    and    are normal and in-plane shear strength along the predefined path respectively. Upon Equation ( 3-1) reaching a value of 1.0, the fracture criterion is satisfied and separation of the nodes are allowed. The failure of the workpiece material is defined by the Tsai-Hill failure criterion which, similar to the nodes separation criterion, is purely-strength based and is governed by Equation ( 3-2).  (     )  (       )  (     )  (      )      (‎3-2) where   ,    and     are the longitudinal, transverse and in-plane shear stresses in the fibre composite respectively, and  ,   and   are the corresponding strength values. Upon Equation ( 3-2) reaching a value of 1.0, the material is considered to have failed. Satisfying the Tsai-Hill criterion along the trimming edge of the composite eventually leads to the release of chip; cutting and thrust forces are recorded upon the node debonding along the trim plane and the workpiece material reaching the Tsai-Hill criterion. It is mentioned that the extent of the Tsai-Hill failure envelope below the trim plane yields a prediction of the sub-surface damage that extends below the trim-plane of the composites. Nayak et al. (2005,b) used a very similar approach to that used by Arola & Ramulu (1997) in numerical modeling of composite orthogonal cutting in which the setup is represented in a two dimensional space with plane stress elements. Similarly, a pre-defined crack path with the same strength based separation criterion as the one used by Nayak et al. (2005,b) and shown in Equation ( 3-1) was implemented along the nodes on the crack path. The main difference in the model by Arola & Ramulu (1997) compared to other numerical models in the literature is in the way the crack path is defined; the chip is considered to form due to the formation of a “primary fracture plane” and a “secondary fracture plane” as shown in Figure  3.2. The primary fracture plane is located along the trim plane and its length is found by analysing the chips generated during the experimental study that was conducted previously. The secondary fracture plane is 36  along the fibre-matrix interface which is the direction along which the chips are released in fibre orientations between 15° and 90° based on the experimental studies. Once debonding of the chip occurs, the two surfaces are allowed to undergo finite sliding; a friction coefficient of 0.4 is defined for this purpose (Arola & Ramulu, 1997; Wang et al., 1995,a).    Figure ‎3.2_Primary and secondary fracture planes used in the numerical model by Arola & Ramulu (1997) Arola & Ramulu (1997), state that the formation of primary fracture plane occurs through the failure of fibres for orientations between 15° and 90° based on the experimental observations. Hence, the normal strength    and shear strength    assigned to the nodes along the primary fracture plane are found by transforming the strength component along and transverse to the fibre axes to the equivalent critical values in the normal and shear direction (XY plane) along the primary fracture plane. For the secondary fracture plane, since the orientation coincides with the local 1-2 axes of the material, the strength properties along these axes is used in Equation ( 3-1) without any transformation (Arola & Ramulu, 1997; Arola et al., 2002; Wang et al., 1995,a). Similar to other numerical studies using the macro-mechanics approach, the cutting forces are in good agreement with the experimental results while the thrust forces are much lower than what has been obtained from the experiments. Another numerical study conducted by Santiuste et al. (2010) assigns Hashin’s failure criterion available in ABAQUS to the workpiece material with both damage initiation and progressive Secondary fracture plane Primary fracture plane ToolθXY37  evolution of damage implemented. In this study, damage is allowed to develop anywhere depending on the stress state of each element rather than having a pre-defined path where damage is limited to. Progressive evolution of damage is considered in both longitudinal and transverse tension and compression. Upon initiation of damage in any mode, the material enters the damage evolution stage at which point the stiffness in each direction (longitudinal, transverse and shear) is degraded based on a damage parameter that varies between 0 and 1 for no damage up to full damage. The ability of the material to sustain damage is determined by the fracture energy per unit cross-sectional area of the material in each failure mode, namely, the longitudinal tension & compression, and transverse tension & compression. Upon saturation of damage, the element is deleted from the mesh. According to the authors, the fracture energy input allows for distinguishing between brittle and ductile behaviour of the material. For CFRP, it is stated that the behaviour is much more brittle than GFRP and that deletion should occur shortly after the onset of damage (Santiuste, Soldani, & Miguelez, 2010; Soldani et al., 2011).  Similar to other studies, the model is created in a two-dimensional space using plane stress elements. The experimental results obtained by Nayak et al. (2005,b) are used to validate the results; it has been shown that the cutting forces are in good agreement with the experiments. Although the predicted thrust forces from the numerical simulations are much lower than the experimental results, the accuracy has been improved compared to other numerical studies as the correct trend of thrust forces with respect to fibre orientation has been obtained.   3.3.2 Three-Dimensional Most of the numerical studies in the field of composites orthogonal cutting is conducted in a two-dimensional space using plane stress elements. A few studies, however, have developed macro-mechanical orthogonal cutting models in three dimensions using brick elements.  Venu Gopala Rao et al. (2008) developed a three-dimensional model where the composite was represented through an EHM with orthotropic properties. The model employs element erosion with the failure of the element governed by the Tsai-Hill failure criterion in three-dimensions. Equation ( 3-2) shown previously represents Tsai-Hill failure criterion in two-dimensions while Equation ( 3-3) is the representation of the Tsai-Hill failure envelope in three-dimensions where    are the normal stress components in the longitudinal, transverse and out-of-plane directions,    ’s are the shear stress tensor acting on an element and                   are the strengths in 38  there three normal and three shear directions respectively. Consistent with other models in the literature, the tool is considered rigid and the contact between the tool and the workpiece uses Coulomb friction law with a friction coefficient of 0.3.   (   )  (   )  (   )      [           ]      [            ]     [           ]  (      )  (      )  (      )      (‎3-3) A comparison of the numerical and experimental results is shown in Figure  3.3 for a cutting tool with a 10° rake angle. Good agreement between the cutting and the thrust forces exits (Venu Gopala Rao et al., 2008).   Figure ‎3.3_Comparison of numerical and experimental results for both cutting forces and thrust forces in orthogonal cutting of CFRP composites with a tool rake of 10° (taken from Venu Gopala Rao et al. 2008) 3.3.2.1 Multidirectional Laminate Modeling In addition to having developed two-dimensional orthogonal cutting models for composite cutting, Santiuste et al. (2011) have also developed a three-dimensional model for the purpose of evaluating the out-of-plane failure and delamination in multidirectional laminates as two-dimensional models fail to capture such mechanisms. The two-dimensional model used Hashin’s failure criterion with damage initiation and subsequent degradation of the elastic properties of the elements based on the fracture energy input for different failure modes followed by element deletion upon reaching saturation energy. In the three-dimensional model however, Hou’s model 39  (Hou, Petrinic, Ruiz, & Hallett, 2000) is used which takes into account the out-of-plane normal and shear stresses in evaluating damage in the through-thickness direction for delamination prediction in addition to longitudinal and transverse damage. Hence the numerical model avoids modeling each ply with a cohesive zone between them to model delamination and instead relies on Equation ( 3-4) which is part of the Hou’s model for predicting out-of-plane damage where         are the through-thickness tensile strength, transverse shear strength and longitudinal shear strength respectively and      is the damage parameter for delamination.   (    )  (     )  (     )        (‎3-4) Upon      reaching a value of 1.0, all the stress out-of-plane normal and shear stress components are reduced to zero. However, if the Hou’s criterion is satisfied for fibre tension or compression mode, all stress components in that element are reduced to zero. An erosion criterion is implemented in the model that erodes the element if any of the principal strain values exceed a user-defined critical value. This model is implemented in ABAQUS Explicit through a user-defined subroutine.  The three-dimensional model created by Santiuste et al. (2011) is used to model both uni-directional and multidirectional laminates and the results are compared with both the two-dimensional models as well as experimental results obtained by Iliescu et al. (2010). The comparison of the matrix damage in the uni-directional case between the two- and three-dimensional models indicate reduced extent of damage in the inner ply compared to the outmost ply; this is said to be due to the existence of the adjacent plies which causes deviation from the plane stress assumption (Santiuste, Miguelez, & Soldani, 2011).  Due to the lack of experimental data on the orthogonal cutting of multidirectional laminates, the numerical results obtained from the multidirectional laminate is compared to the summation of average cutting forces required for each uni-directional ply which is similar to the approach adopted by Wang et al. (1995,b) in the experimental evaluation of cutting forces in a multidirectional laminate. It was shown that the numerical results correspond accurately to the estimated cutting forces. 40  Delamination damage was shown to extend ahead of the tool and was more extensive between plies with large difference in fibre orientation, such as 45° & -45° and 0° & 90° fibre orientations due to the out-of-plane shear stresses. An example of the delamination prediction is presented in Figure  3.4.       Figure ‎3.4_Delamination damage (shown in red) in orthogonal cutting of multidirectional laminate (taken from Santiuste et al. 2011) As stated, intralaminar damage was found to be more extensive in the outer surface plies compared to the interior plies; delamination damage also tends to be more extensive in between plies closer to the surface with maximum delamination predicted between the        plies in the                 stack and between the      plies in the                 stack (Santiuste et al., 2011). 3.4 Micro-Mechanical Models  In modeling composites using the micro-mechanics approach, the fibre, the matrix and the interface between the two material phases present in FRP composites is considered. Some of the numerical simulations which used the micro-mechanics approach are summarized here along with the material models used to model each phase of the composite.  Nayak et al. (2005,b) developed a micro-mechanics two-dimensional model for orthogonal cutting of GFRP alongside the macro-mechanics model described in the previous section. In the micro-mechanical model, a single fibre is modeled surrounded by the matrix material with the tool initially in contact with the fibre as shown in Figure  3.5. The boundary between the fibre and the matrix is assigned a node separation criterion which similar to macro-mechanics model, is 41  governed by Equation ( 3-1) with the normal (    and shear strength      of the interfaces set to 160 and 34 MPa, respectively. An example of the node separation between the fibre and the matrix phase is shown in Figure  3.6. The fracture of the fibre and the formation of the chip are characterized by the principal stresses in the fibre phase reaching its tensile or compressive strength limit; hence no progressive damage of the fibre or matrix is considered in this model. The machining forces are recorded once the fibre has fractured and a chip has formed (Nayak et al., 2005,b).   Figure ‎3.5_Schematic view of the micro mechanics orthogonal model for orthogonal cutting of GFRP developed by Nayak et al. (2005,b) Using this method, Nayak et al. (2005,b) obtained good agreement between the experimental results and prediction of both the cutting forces and thrust forces. The predicted trend of force variation versus the fibre orientation matches closely with the experimental data for both thrust and cutting forces; hence, this study shows that the shortcomings of the macro mechanics approach in predicting the thrust forces can be overcome. An example of the numerically evaluated cutting and thrust forces and their comparison with experimental results is shown in Figure  3.7. 42   Figure ‎3.6_Micro-mechanical model for 75° fibre orientation showing separation between the fibre and the matrix phase (taken from Nayak et al. 2005,b)   Figure ‎3.7_Comparison of numerical and experimental cutting and thrust forces (taken from Nayak et al. 2005,b) Venu Gopala Rao et al. (2007,b) conduct a numerical study of CFRP and GFRP orthogonal cutting using a micro-mechanical model. The model developed in this study is more complex than the one developed by Nayak et al. (2005,b) in that multiple repetitions of the fibre and matrix phase are modeled as opposed to a single fibre. The interface between the fibre and the matrix phase contains zero thickness cohesive elements with a traction separation law with stiffness degradation upon damage initiation which allows for separation once the interfacial fracture energy is exceeded. The matrix is also modeled with damage initiation and evolution through isotropic hardening and stiffness degradation to mimic the formation of voids and micro-cracks. Similar to Nayak et al. (2005,b) however, the chip formation and fibre fracture is defined 43  by the fibre principal stresses reaching the tensile or the compressive limit. The computed cutting and thrust forces from the numerical models is shown to agree well with the experimental results shown in Figure  2.7. It must be noted that both the cutting and the thrust forces in both the experiments and the numerical simulations were lower for CFRP than for GFRP.  The cohesive zone used in this model takes into account both mode I (normal) and mode II (shear) openings. A fracture energy and a strength value is associated with each opening mode as shown in Figure  3.8. The area under the plot defined by    (with       ) represents the energy per unit cross-sectional area of the element needed to damage saturation (at complete separation) for a particular opening mode and    and    are the displacements at which damage initiates and saturates, respectively. In order to account for mixed mode opening however, damage initiation occurs upon satisfying Equation ( 3-5) while full damage saturation under mixed mode condition is achieved upon satisfying Equation ( 3-6) where    and    are the fracture energies consumed in the normal and shear modes respectively.   (    )  (     )    (‎3-5)   (    )  (     )      (‎3-6)  Figure ‎3.8_Transction-separation response for the interfacial cohesive elements for each opening mode  Both Venu Gopala Rao et al. (2007,b) and Nayak et al. (2005,b) show the stress distribution in the fibre phase to be composed of compression at the point of contact with the tool and tensile 44  stresses immediately outside the compression zone all the way to the back of the fibre. Nayak et al. (2005,b) found that for all fibre orientations fracture was induced as a result of tensile stresses reaching the fibre tensile strength limit at the region opposite to the tool-workpiece contact. It was also stated that the maximum in-plane tensile stresses induced in the fibre is along the fibre axes which in turn implies fracture along the transverse axes of the fibre. This statement is in agreement with experimental observations. Similarly, Venu Gopala Rao et al. (2007,b) found that for GFRP fibres the fibre fracture mode is initiated as a result of tensile bending stresses reaching the fibre tensile strength limit. Even though the compressive stresses in the contact zone are of the same order magnitude as the tensile stresses, the compressive strength of GFRP fibres is said to be five times greater than its tensile strength. For CFRP composites however, Venu Gopala Rao et al. (2007,b) states that fibre fracture is induced as a result of both tensile failure of the fibres as well as compressive crushing at the contact region (Nayak et al., 2005,b; Venu Gopala Rao et al., 2007,b).    Dandekar & Shin (2008) developed a multiphase numerical model for orthogonal cutting of CFRP and GFRP and compares the cutting forces and spread of damage with the experiments conducted alongside this study. The developed model includes damage initiation and evolution in the fibre phase in addition to damage evolution in the matrix phase and the interface between the fibre and the matrix. The elastic modulus of the fibre phase is degraded as shown in Equation ( 3-7). The damage parameter   is a function of the strain energy release rate in the fibre and the Weibull distribution parameters which accounts for the statistical variance in the strength of the fibres (Dandekar & Shin, 2008).              (‎3-7) The predicted cutting and thrust forces as well as the extent of the fibre-matrix debonding into the depth of the material are in good agreement with the experimental results obtained by Nayak et al. (2005,a) which were used for validation purposes for the GFRP composites. It was successfully shown through the numerical simulation that with increasing fibre orientation from 45° to 90°, subsurface damage through the fibre matrix debonding extends to a greater depth into the specimen. The depth of the predicted damage compares well with experimental observation. Fibre compressive failure on the front side where the tool is in contact with the fibre along with tensile rupture on the back face of the fibre is shown through the numerical simulation. This 45  finding is in line with the numerical results obtained by other literatures (Venu Gopala Rao et al., 2007,a; Dandekar & Shin, 2008; Nayak et al., 2005,a).   In another numerical study by Abena et al. (2015) where multiple material phases has been considered, the fibre phase has been modeled such that the material remains fully elastic prior to failure with fracture in the fibre defined based on the maximum principal stress criterion in order to simulate its brittle nature. The cohesive zone between the fibre and the matrix phase however is assigned a finite thickness rather than the commonly zero thickness elements used in other literature. This is done in order to accommodate compressive failure of the cohesive zone. Damage in the matrix phase is accounted for through an elastic-plastic response (Abena et al., 2015). In the study by Iliescu et al. (2010), a Discrete Element Method (DEM) is used rather than the conventional finite element method approach used in most studies. A micro-mechanical representation of the material is created with particles with mass and interaction properties between them. In this approach, at each time step, the acceleration, velocity and displacement of each particle is calculated through the Newton’s law and the contact and interactions properties defined between the particles. Predicted cutting forces and thrust forces has been shown to agree well with the experimental results for fibre orientations of 0°, 45°, 90° and 135° (Iliescu et al., 2010).  3.5 Summary  Numerical studies of orthogonal cutting of composites is a relatively new and underdeveloped field. There exists limited numerical representation of the orthogonal cutting of composites in the literature where it has been shown that it is possible to obtain a reasonable prediction of the cutting forces from numerical models, although most of the numerical representations only go as far as chip initiation and in many cases fail to show the complete chip formation process. Also, most literatures do not provide the numerical results in the form of a cutting force history plot. The damage models used in most numerical studies is strength based where elements are deleted or degraded upon satisfying a stress-based criterion without any ductility or ability to dissipate energy past their failure strength. A list of the numerical studies conducted is compiled in Table  3-1. 46  Table ‎3-1_List of numerical models on orthogonal cutting of composites Author Micro Mechanics (multiphase) Macro Mechanics (EHM) Analytical approach (Arola & Ramulu, 1997)    (Pwu & Hocheng, 1998)    (Mahdi & Zhang, 2001)    (Zitoune, Collombet, Lachaud, Piquet, & Pasquet, 2005)    (Nayak, Bhatnagar, & Mahajan, 2005,b)    (Zhang, Zhang, & Wang, 2006)    (Venu Gopala Rao, Mahajan, & Bhatnagar, 2007,b)    (Venu Gopala Rao, Mahajan, & Bhatnagar, 2008)   (3D model)  (Dandekar & Shin, 2008)    (Mkaddem, Demirci, & El Mansori, 2008)    (Lasri, Nouari, & Mansori, 2009)    (Jahromi & Bahr, 2010)    (Santiuste, Soldani, & Miguelez, 2010)    (Santiuste, Miguelez, & Soldani, 2011)   (3D model)  (Iliescu, Gehin, Iordanoff, Girot, & Gutierrez, 2010)  (discrete element method)   (Abena, Leung Soo, & Essa, 2015)    47   Composite Material Models in ABAQUS and LS-Dyna  Chapter 44.1 Introduction  ABAQUS and LS-Dyna are the two numerical codes selected for evaluating orthogonal cutting of composites. As evident form the review of the literature in the previous chapter, ABAQUS Explicit is the more popular code used for the purpose of composite orthogonal cutting simulation for both micro- and macro-mechanical models. LS-Dyna on the other hand is known for its explicit capabilities and is widely used in the Composite Research Network (CRN) at the University of British Columbia for the purpose of composite damage modeling.  In this chapter, the material models used in the numerical codes, ABAQUS and LS-Dyna, to model orthogonal cutting of composites are expressed through uniaxial loading of single elements. The material models presented here include the Hashin’s damage model for fibre reinforced composites available in ABAQUS as a built-in material model as well as MAT_054-Enhanced Composite Damage model available in LS-Dyna. Both material models handle composite with anisotropic material properties. The two models differ significantly in how damage is evaluated in each principal direction as well as the way in which they treat the interactions between different modes of loading. The damage model in ABAQUS relies on fracture energies whereas in LS-Dyna, the MAT_054 relies purely on strain in the longitudinal (fibre) and transverse (matrix) direction without considering any interaction between the loads in the longitudinal and transverse direction. The numerical models in composites rely on element erosion scheme to deal with failed or highly distorted elements; the erosion criteria used in each material model is also presented in this chapter.   4.2 Hashin’s‎Damage Model in ABAQUS   The numerical solver ABAQUS offers a built-in material model for composite damage that is based on Hashin’s theory. This material model allows for prediction of damage initiation as well as damage evolution in the material based on a fracture energy which is a user input. Hashin’s constitutive model in ABAQUS is limited to two- and three-dimensional shell and membrane elements governed by plane stress formulation where all the stress components are within the 1 and 2 principal planes of the composite, namely    ,    , and     which are the normal stress 48  components in the longitudinal and transverse directions and the in-plane shear stress respectively. Hashin’s damage model considers damage initiation in the following four modes:   Fibre (longitudinal) tension       Fibre (longitudinal) compression      Matrix (transverse) tension       Matrix (transverse) compression     Upon damage initiating in any one mode, its evolution leads to stiffness degradation in that mode until complete saturation of damage is reached, Details specific to the initiation and evolution of damage are provided in the following sections.  4.2.1 Damage Initiation   As mentioned, initiation of damage can occur in any one of the four modes of longitudinal tension or compression and transverse tension or compression. Each one of the damage initiation modes is governed by the criteria presented in Equation ( 4-1). For damage to initiate in any of the stated modes, the initiation criterion must be satisfied for that particular mode. When the initiation criterion reaches a value of 1.0 for a given mode and damage initiates and leads to the subsequent degradation of the material properties and evolution of the damage in that mode.     {            (      )     (      )     (      )  (       )     (        )  *(      )   +       (       )     (‎4-1)   and   denote the longitudinal and transverse strength values with subscripts   and   referring to the compressive and tensile values of these quantities.    and     denote the in-plane and transverse shear strength respectively. The normal and shear stresses     ,     , and      with the subscript   denote the effective stresses which represent the stresses acting on an equivalent undamaged material subjected to the same strain as the damaged material; the effective is computed based on Equation ( 4-2).  49          (‎4-2) where  is computed from the damage parameters once damage has been initiated in any mode; the damage parameters and the matrix  are derived as shown in Equation ( 4-3) where       are the damage parameters computed for each of the longitudinal, transverse and shear modes.     [                           ]       (‎4-3) The damage parameters are initially zero and hence the value of parameter  is unity which in turn suggests that the effective stresses are equal to the stresses since no damage exists. As soon as damage initiates in a given mode, damage evolution begins and the parameter   is computed (      ).  4.2.2 Damage Evolution  The damage evolution in Hashin’s constitutive model is a generalization of that implemented for capturing the response of a cohesive zone described in section  3.4. Similar to the inputs required for the cohesive zone, Hashin’s model requires a fracture energy input for each of the stated damage modes repeated here:   Fibre (longitudinal) tension       Fibre (longitudinal) compression      Matrix (transverse) tension       Matrix (transverse) compression     The fracture energy is a measure of energy required per unit cross sectional area of the element that is perpendicular to the loading direction to yield complete degradation of the properties in that mode; upon satisfying any of the initiation criteria in Equation ( 4-1), evolution of damage in that mode takes place according to the stress displacement plot shown in Figure  4.1 where the stiffness of the material is degraded linearly. 50   Figure ‎4.1_Damage evolution stress-displacement plot for Hashin's damage model In Figure  4.1,    is the fracture energy input for any of the four stated damage modes and       and       are the equivalent stress and displacement at the onset of damage and       is the displacement at full damage for damage mode  .  The equivalent displacements and stresses      and      for the longitudinal tension or compression modes (    and    ) are calculated from the stresses and strains in the longitudinal 1-direction whereas for the transverse damage modes (    and    ) the equivalent stresses and displacements are calculated from the stresses and displacements in the transverse 2-direction and the shear 12-directions. The displacement in every direction is calculated by multiplying the strain in that direction (on plane) by a characteristic length    which is equal to the square root of the area of the element as represented in Equation ( 4-4).             (‎4-4) where    is the direction of displacement (longitudinal, transverse or shear). The damage parameters    which are computed for the longitudinal, transverse and shear modes are a measure of the degradation of the material stiffness in each direction and are calculated based on Equation ( 4-5) for the longitudinal and transverse directions. For the shear direction, the damage parameter is calculated from the damage parameter computed for the principal directions as shown in Equation ( 4-6).            (          )      (           )               (‎4-5) 51                                         (‎4-6) Complete degradation of the element properties in the longitudinal, transverse or shear direction is attained when the damage parameter in each direction reaches the maximum value of 1.0, which is an indication that      has reached the value of      .      4.2.3 Element Erosion in ABAQUS Explicit As mentioned, in modeling of composite orthogonal cutting, element erosion scheme is used whereby fully damaged elements which no longer contribute to the stiffness and strength of the workpiece are removed from the mesh. ABAQUS Explicit provides such capability; in ABAQUS Explicit using Hashin’s constitutive model, an element is considered to have failed when the damage parameter in the longitudinal (fibre) direction reaches 1.0 or a reduced value based on the user input as ABAQUS Explicit allows for removal of the element at a user-defined damage parameter of less than 1.0 which is the default value. Therefore, Hashin’s constitutive model in ABAQUS Explicit requires the element to reach complete saturation in longitudinal tension or compression before it removes the element; as a result, the material model is very suitable for use in orthogonal cutting of composites since based on the experimental observations reviewed in Review, fracturing of the fibre results in the formation of chips and Hashin’s model in ABAQUS requires complete saturation in the fibre direction prior to removing the element.  A parameter in ABAQUS Explicit defined as Max Degradation gives users the ability to input a maximum damage parameter value at which the user intends to fail and remove the element; this user input overrides the default value of 1.0 for the damage parameter    at maximum degradation and results in the element deleting before full saturation and complete degradation of its properties. The use of this parameter in the will be further explained in Chapter  Chapter 7.  4.2.4 Single Element Examples  In this section, a few examples of uniaxial single element test are provided to better illustrate the material behaviour and response. The numerical setup consists of a single (one by one) square element in the two-dimensional plane with element type CPS4R in ABAQUS Explicit which is a reduced integrated four-node bilinear element with plane stress formulation. Figure  4.2 provides a visual representation of the single element setup with the imposed boundary conditions. As the 52  setup is meant to stress the element in a uniaxial manner, the boundary conditions imposed are such that the element is free to strain in the global X direction while the two nodes on the top are constrained in the global Y direction. Velocity boundary condition is imposed on the two bottom nodes of the element in the positive or negative global Y direction in order to test the element in compression or tension. The material orientation (local 1-2 direction) is adjusted according to the direction in which the material is intended to be tested.    Figure ‎4.2_Single element setup used to evaluate the material response for Hashin's constitutive model The material properties used are taken from Santiuste et al. (2010) which are typical of AS4/3501-6 CFRP composite. The elastic and the strength properties are listed in Table  4-1.          XY53  Table ‎4-1_Mechanical properties of AS4/3501-6 CFRP composite Properties Values  Elastic Modulus 1 (E1) 126,000 MPa Elastic Modulus 2 (E2) 1,000 MPa In Plane Shear Modulus (G12) 6,600 MPa Major/Minor Poisson’s ratio (ν12/ν21) 0.28/ 0.0244 Mass density (ρ) 0.0017       Fibre Compressive Strength (11) 1,480 MPa Fibre Tensile Strength (11) 1,950 MPa Transverse Compressive Strength (22) 200 MPa Transverse Tensile Strength (22) 48 MPa   For the purpose of testing the single element, the fracture energy values for the fibre tension and compression are both taken to be             (    ). It must be noted that this value is not particular to the material listed in Table  4-1, but rather a value in the same order magnitude as the fracture energy for a typical carbon fibre epoxy composite which the material in Table  4-1 represents. Also, this is merely meant to illustrate the response of the material and the accuracy of this value is irrelevant to this section.  In the first example, the material is tested in fibre compression; the nodes are assigned a velocity of         in the direction shown and the local axes of the material are orientated as illustrated in Figure  4.3. The expected values of displacement at onset of damage and failure as well as the energy dissipated by the element are summarized in Table  4-2.    54   Figure ‎4.3_Single element setup for testing fibre compression with fracture energy of 150 kJ/m2   Table ‎4-2_Expected outcome of the single element test for fibre compression mode Properties  Values  Stress at damage onset       1,480 MPa Displacement at damage onset      0.0117 mm Displacement at complete saturation      0.2027 mm  Energy to reach full degradation  150 mJ  The stress displacement plot obtained from this test is as shown in Figure  4.4. As it can be seen, the results are identical to what had been calculated and expected. The element is eroded at a displacement of 0.2025 mm which is the displacement at which the damage parameter reaches a value of 1.0 and complete degradation of the element in longitudinal compression has been attained. XY1255   Figure ‎4.4_Single element longitudinal compression damage initiation and evolution A similar example for transverse (matrix) tension failure mode is also performed. The stress-displacement response in this loading scenario is also similar to the longitudinal compression mode shown in Figure  4.4 with different values for the damage onset and displacement at complete degradation. The major difference associated with damage evolution in the transverse direction compared to the longitudinal direction is that the saturation of the material in the transverse direction does not cause the erosion of the element since ABAQUS Explicit requires failure of fibre in either tension or compression in order to consider the element failed and hence its removal from the mesh.   To better illustrate how the material model captures the damage in tension and compression, in the next example presented, the single element is loaded in the local 1-direction in compression first, followed by tension and then compression again up to failure and deletion. The material properties are identical to the previous case and are listed in Table  4-1 with fracture energy values 00.10.20.30.40.50.60.70.80.91-1500-1250-1000-750-500-25000 0.05 0.1 0.15 0.2 0.25 0.3Fibre Compression Damage Parameter "d"Longitudinal Stress (MPa)End node displacement (mm)Stress-Displacement ForLongitudinal (Fibre) CompressionFibre Compression DamageParameter "d"XY12Damage onset Element erosion56  of          in both tension and compression. Table  4-3 along with Figure  4.5 summarizes the three stages in the single element run. Initially, the element is compressed.    Figure ‎4.5_Single element run with multiple steps  Table ‎4-3_Summary of the steps in the single element run for testing the element in longitudinal tension and compression Stages Time      Duration      Velocity (    ) Start Finish Initial Compression  0 0.5 0.5 0.2 Tension 0.5 1.5 1.0 -0.15 Final Compression 1.5 3.5 2.0 0.2  The result from the test summarized in Figure  4.5 is plotted in Figure  4.6. Initially, the material is displaced in compression to a displacement of 0.1 mm which is well past its damage onset displacement but less than the displacement at full degradation of the element in longitudinal compression as shown by the fine dotted orange line (OAB). The element is then subjected to tension by stretching it for 0.15 mm. As it can be seen in the plot, the element is unloaded on a reduced stiffness (BO) due to undergoing damage in compression. However, once it reaches its original position at point “O”, the tensile stresses develop on the initial longitudinal stiffness (OC) as the damage parameter     is initially zero; similar to the previous step, at point “C” the XY121212Initial compressionFrom t=0 to t=0.5 msTension From t=0.5 ms to t=1.5 msFinal compressionFrom t=1.5 ms to t=3.5 ms57  element is displaced beyond its damage onset in tension and displacement occurs along “CD”. In the “Final Compression” stage, the element is compressed from a net extension of 0.05 mm at “D” to until full degradation and erosion of the element at “E” (DOBE). The element is unloaded on a reduced stiffness in tension (DO) and reloaded in compression on the previously reduced stiffness due to damage in compression (OB). The damage parameter is calculated at each step based on the displacement and compared with the previously stored damage parameter; the damage parameter is updated once it exceeds the stored value, at which point the stresses reduce again until full damage has been achieved as shown by the line “BE”. As expected, the element fails at the calculated displacement shown in Table  4-2. The damage parameter for fibre compression has also been plotted in Figure  4.6 on secondary axes to show the development of damage at each stage of the test.   Figure ‎4.6_Single element results for the compression-tension-compression load element 00.10.20.30.40.50.60.70.80.91-1500-1000-5000500100015002000-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3Fibre Compression Damage ParameterLongitudinal Stress (MPa)End node displacement (mm)Initial CompressionFinal CompressionTensionFibre Compression Damage Parameter (initial compression)Fibre Compression Damage Parameter (final compression)Fibre Compression Damage Parameter (tension)Element erosionOABCDE58  4.3 LS-Dyna MAT_054 Enhanced Composite Damage  The Enhanced Composite Damage model available in LS-Dyna under MAT_054 is an elastic perfectly plastic orthotropic material model and is restricted to thin shells only. The material model requires as input the usual moduli of elasticity and Poisson’s ratios in the principal 1- and 2- directions. For damage and erosion purposes, the user has the option to input yield/failure strengths and strain-to-failure values for longitudinal tension, longitudinal compression, transverse tension, and transverse compression and in-plane shear. The material behaviour in this model depends on the set of damage parameters defined as inputs; the response in each direction of loading is found to be dependent on what set of parameters has been defined for that mode. In the following, the material behaviour is explained more thoroughly.   4.3.1 Strength Parameter  As mentioned, strength for the longitudinal and transverse loading in both tension and compression as well as strength in shear can be assigned to the material model. Assigning a strength value in each direction results in the yielding of the material in that direction upon reaching the specified strength value. If no strength value is assigned, the material integration point behaves elastically.  The post-yield response of the material model depends on the damage parameters defined for each mode, which is the strain-to-failure of the elements as well as the direction of loading. The next section explains this matter.  4.3.2 Strain-to-Failure Parameter The material model gives the user the option to input a value for the strain at which the material fails and is therefore removed from the mesh. In the case where a strain-to-failure is associated with a direction along with a yield strength, the material undergoes a perfectly plastic response after the yield strength up to the strain-to-failure defined for that mode. The strain-to-failure value in the transverse direction is limited to a maximum value of 1.0 and can be overwritten by inputting a lower value. In the longitudinal direction however, despite the user manual stating a maximum allowable value of 1.0 for the strain-to-failure, it was found that values exceeding this limit are also permitted. The element is eroded once the strain-to-failure in any direction is reached.   59  4.3.3 Single Element Runs in LS-Dyna  In this section, uniaxial single element tests are conducted in LS-Dyna to illustrate the material behaviour and response for MAT_054. The setup of the single element in terms of size and boundary conditions is identical to that shown in Figure  4.2. Shell element Type 16 “Fully Integrated Shell” in LS-Dyna with a unit thickness is used. The end nodes are assigned a displacement boundary condition which varies linearly with time whereby the displacement is applied gradually over a certain period of time with at a constant rate. The elastic properties of the material are as shown in Table  4-1.    In the first example, the yield strain and the strain-to-failure value for longitudinal tensile and compressive loading are as listed in Table  4-4. The yield strain is calculated by dividing the strength in each direction by the longitudinal stiffness (Table  4-1). Strain-to-failure values are calculated such that the fracture energy for the single element equal to 150      in the longitudinal tension or compression. The strain-to-failure can also be inputted for the transverse direction as well, however, the material model only allows a single value to be assigned to the transverse direction which is applied to both tensile and compressive loads. It must be noted that the values for the fracture energies and hence strain-to-failures are not accurate and are merely for demonstrating the material behaviour in using MAT_054 in LS-Dyna.  Table ‎4-4_Strain-to-failure parameters in LS-Dyna Loading mode Yield strain Strain-to-failure Equivalent fracture energy        (    ) Longitudinal (Fibre) Tension 0.0155 0.084 150 Longitudinal (Fibre) Compression 0.0117 0.107 150  In the first example, the element is displaced along its principal (longitudinal) 1-direction in tension. A total displacement of 0.2 mm over 2.0 seconds is applied (i.e. a velocity of 0.1 mm/ms). The material response is as shown in Figure  4.7. The material is linearly elastic up to 1950 MPa at which point it yields; the element undergoes perfectly plastic deformation until it reaches a strain of 0.084 which is the defined strain-to-failure for longitudinal tensile loading. 60  Note that the strains used by the material model are the true (logarithmic) strains as shown in Equation ( 4-7) where    and    are the final and initial length of the element respectively. Therefore a strain of 0.084 corresponds to a final length of 1.0876 for an element with an initial length of 1.0 mm hence yielding a net displacement of 0.0876 mm.         (    ) (‎4-7)   Figure ‎4.7_Stress-displacement for single element in tension In the next example, the response of the element is shown in a scenario whereby the element is loaded in tension and unloaded before failure and reloaded in compression up to failure in two stages (Figure  4.8). The element is first pulled 0.05 mm (tension) in 2.0 ms as shown by the green line (OAB); the element yields at the input yield stress and deforms plastically up to a displacement of 0.05 mm. The end nodes are then displaced in the opposite direction as represented by the red line (BCDE). As indicated, the element is unloaded on its initial stiffness as oppose to Hashin’s constitutive model in ABAQUS where the unloading of the element occurs on a reduced damaged stiffness. Hence using MAT_054 in LS-Dyna, the element has some residual displacement at zero stress. The reloading in compression occurs on the same 02004006008001000120014001600180020000 0.02 0.04 0.06 0.08 0.1Stress (MPa)Displacement (mm)XY12Element deleted at displacement of 0.0876 mm (strain of 0.084) 61  initial stiffness and yielding in compression occurs at 1,480 MPa yield stress. As stated, the element erosion is controlled purely by the strain and the element is deleted at a compression strain of 0.107 (Point E).    Figure ‎4.8_Stress-displacement plot for a single element in LS-Dyna in a tension-compression cycle          -1,500.00-1,000.00-500.000.00500.001,000.001,500.002,000.00-0.125 -0.1 -0.075 -0.05 -0.025 0 0.025 0.05Stress (MPa)Displacement (mm) OA BElement deleted at displacement of 0.1014 mm compression  (strain of -0.107)  CDE62   Orthogonal Cutting of Metals: Numerical Modeling   Chapter 55.1 Introduction  The main objective of this study is to evaluate the numerical modeling of composite machining and to assess the feasibility and the capability of the available numerical tools such as ABAQUS and LS-Dyna for this purpose. Few literature studies exist on the machining of composites, most of which conduct experimental studies with very few focusing on the numerical modeling of composite machining; amongst the limited numerical studies, ABAQUS is the most commonly used numerical platform for the purpose of modeling machining of composites. There have been some numerical studies of orthogonal cutting and drilling of composites using ABAQUS (Isbilir & Ghassemieh, 2014; Santiuste et al., 2010).  In this section, the numerical approaches used in this study to model the orthogonal cutting of composites are described in detail. Orthogonal cutting models have been set up in both ABAQUS and LS-Dyna software. First, orthogonal cutting of metals using an element deletion approach is set up in LS-Dyna and the results are compared with a validated model previously developed in ABAQUS by the Manufacturing Automation Group in the Mechanical Engineering Department at the University of British Columbia. Subsequently, composite orthogonal cutting models are set up in both ABAQUS and LS-Dyna. The results are compared to a limited number of available data in the literature.    Amongst the machining operations, orthogonal cutting of composites is chosen as the most fundamental approach. As previously explained in the literature review part of this thesis, the aim is to apply the same framework as in metal machining where a geometric and kinematic transformation of the orthogonal cutting model is used to predict the cutting force history plots for other complex machining operations such as drilling and milling. Zitoune et al. (2005) and Bhatnagar et al. (1995) also performed experimental investigation of orthogonal cutting of composites and specifically mentioned as taking this approach in order to understand the cutting mechanisms in the drilling and other machining operations in composites (Bhatnagar et al., 1995; Zitoune et al., 2005).  63  5.2 Orthogonal Cutting of Metals in ABAQUS The metal orthogonal cutting model developed in ABAQUS relies on an adaptive meshing scheme or Arbitrary Lagrangian Eulerian (ALE) approach in order to deal with highly distorted elements; no elements are eroded during the numerical run and instead, the mesh is regenerated after each step. Therefore, the mesh is not attached to the underlying material and is free to move independently in order to maintain a high quality mesh and prevent excessive distortion of elements.      5.2.1 Johnson-Cook Constitutive Model  In machining of metals, the chip formation is based on the shear yielding and the plastic work in the material. For this purpose, the Johnson-Cook constitutive material model implemented in numerical platforms such as ABAQUS and LS-Dyna is used. In this constitutive model, non-linear strain hardening, strain rate and thermal softening effects are taken into consideration in the equation of the effective yield stress as given by Equation ( 5-1). Table  5-1 lists the Johnson-Cook parameters as well as the elastic constants for brass.      (    ̅  )       ̇  (     ) (‎5-1) Table ‎5-1_Input parameters and material properties for Brass Elastic Modulus (E) 110,000 MPa Shear Modulus (G) 41,350 MPa Poisson’s ratio (ν) 0.33 Mass density (ρ) 0.0085       A (Johnson Cook input) 90.0 MPa B (Johnson Cook input) 404.0 MPa n (Johnson Cook input) 0.42 c (Johnson Cook input) 0.009 m (Johnson Cook input) 1.68 Melt Temperature  916 °C Specific Heat 377 mJ/(g. °C)  The term  ̅ and    in Equation ( 5-1) are the effective plastic strain and the homologous temperature terms respectively and  ̇  is the effective plastic strain rate.  64  5.2.2 Finite Element Model  In this section, the different parts of the orthogonal cutting model are described in details. The model consists of two main components: the workpiece and the cutting tool. Each component of the model has its specific geometric features and is assigned certain properties and the interaction between the two parts is made possible by defining a contact interaction between them.   5.2.2.1 Brass Workpiece  An orthogonal cutting model of brass has been developed in ABAQUS by the Manufacturing Automation Group at the University of British Columbia which is used as a benchmark model for the purpose of this study. The model is developed using ABAQUS Explicit. The size of the specimen is 270    by 85    with a cutting depth of 5   . Element type CPE4RT (linear quadrilateral plane strain element) in ABAQUS is used to discretize the workpiece.The model is run for both unit thickness and 50    thickness values for comparison purposes. The elements are under-integrated with hourglass control enabled. The material properties assigned to the workpiece section are listed in Table  5-1. A setup of the descretized model is shown in Figure  5.1. As it can be seen, the mesh in the vicinity of the contact region between the tool and the workpiece material is very refined and the mesh size in this region is around 0.5   .   Arbitrary Lagrangian Eulerian (ALE) adaptive meshing has been used in this model where the workpiece is re-meshed throughout the analysis such that the mesh remains refined and in the same order of size in the region around the tool and the workpiece and coarse elsewhere where no action is taking place. The ALE method allows the mesh to move independently of the material. It must be noted that the element deletion criteria is not adopted and the chip formation is purely based on the inelastic deformation of the material where the distortion of the elements and the large deformations are controlled by the adaptive meshing scheme.  65   Figure ‎5.1_Orthogonal cutting of brass, setup in ABAQUS Explicit 5.2.2.2 Tool Material  The cutting tool is carbide with geometric features as summarized in Table  5-2 and the tool illustrated in Figure  5.2. The tool is assigned rigid constraints and therefore does not deform; this is very common practice in both metal and composite orthogonal cutting. Similar to the workpiece, the mesh density in the region around the tool nose where there is contact between the tool and the workpiece has a refined mesh in the order of 1   . Despite the tool being considered rigid, the mesh density in the contact region must be in the same order as the workpiece in order to be able to accurately model the contact and prevent penetration of the parts into each other. The mesh away from the tool nose is coarse for increased computational efficiency.  Table ‎5-2_Geometric features of the cutting tool α: rake angle 5°  γ: relief angle 7°  r: tool nose radius 3.7    XYVV66   Figure ‎5.2_Cutting tool in ABAQUS 5.2.2.3 Contact and Boundary Conditions  In the assembly module in ABAQUS, the tool and the workpiece are brought close to each other; the 5    cutting depth is defined on the workpiece and the cutting tool is placed such that it fits in a 5    deep slot with a radius that matches the radius of the tool insert. The cutting tool is fixed in space in all directions. The base of the workpiece is fixed in the vertical direction and the two vertical sides are assigned a velocity of 420 mm/s (25.2 m/min) in the positive X-direction as shown by the arrows in Figure  5.1.    Contact between the two surfaces of the workpiece and the cutting tool is enforced using the penalty formulation. Tangential (frictional) behaviour between the two surfaces is modeled with a friction coefficient of 0.15.    αγr67  5.2.3 ABAQUS Results  The cutting forces obtained from the model are illustrated in Figure  5.3. As previously mentioned, two models are tested with the cutting depths and the cutting widths of 5 and 50    respectively. Since we are interested in the cutting forces per unit width of the material, an identical model with the plane strain thickness set to 1.0 mm was also run with the corresponding results shown on a secondary axis in the plot.   As it can be seen from the plot in Figure  5.3, up to a cutting force of about 0.12 N (or 2.4 N/mm), the force-displacement is linear. The linear response region is isolated and shown in detail in the insert on the same plot. This suggests the material remains linearly elastic and the load rises linearly with displacement of the workpiece up to a cutting force of 0.12 N at which point the material yields. From that point on, the material strain hardens as governed by the Johnson-Cook relation and the load rises continuously. Once a chip has fully developed through the yielding of the material along a shear plane, the cutting forces reach a steady state value of 0.38 N (or 7.6 N/mm); this in turn yields a cutting coefficient of 1520 MPa. The cutting coefficient is the cutting force required per unit area of the cut with the area defined as the width of the cut multiplied by the depth of the cut as the cutting force is directly proportional to the depth and the width of the material being cut.  As we would expect, the plots for the two thickness values match with the magnitude of the force for the unit thickness model being twenty times more than the model with 50    (= 0.050 mm) thickness as the cutting forces are directly proportional to the depth and width of the cut . The slight difference in the oscillation and the noise present in the two plots, especially past the 0.025 mm displacement where a chip has fully formed and both the material and the geometries are well in the non-linear range, can be attributed to the characteristics of the explicit analysis yielding a very slight difference in the two plots. As the model is in two dimensions only, ideally dividing the magnitude of the cutting forces by the 50    thickness should yield the cutting forces per unit width.   68   Figure ‎5.3_Cutting force from orthogonal cutting simulation of brass in ABAQUS The formation of the chip through yielding of the material along the shear plane is shown in Figure  5.4. The figure shows the shear stresses at 0.026 mm workpiece displacement where the shear plane and the chip have fully developed and the cutting forces are almost reaching the steady state. This shear plane is an inclined plane through which the material undergoes shear deformation and the chip is formed.  01234567800.050.10.150.20.250.30.350.40 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08Cutting Force (N/mm)Cutting Force (N)Workpiece Displacement (mm)thickness = 0.050 mm thickness = 1.0 mm0123400.040.080.120.160.20 0.001 0.002Cutting Force (N/mm)Cutting Force (N)Workpiece Displacement (mm)69   Figure ‎5.4_ Chip formation through yielding along a shear plane at 0.026 mm of workpiece displacement (shear stresses shown) in orthogonal cutting simulation of brass  5.3 Orthogonal Cutting of Metals in LS-Dyna As this study aims to evaluate the orthogonal cutting of composites with LS-Dyna, it was considered important to first attempt the orthogonal cutting of metals and compare the results with the validated model from ABAQUS in order to gain confidence with the ability of this code to model such events. Therefore, a model is created in LS-Dyna where nearly all the features of interest are implemented with certain differences as explained in this section.   5.3.1 Johnson-Cook Constitutive model As stated previously, the ABAQUS model relies on adaptive meshing algorithm (Arbitrary Lagrangian Eulerian adaptive meshing) in order to cope with the large element distortions that occur. In the orthogonal cutting of composites however, the approach taken in the numerical analysis involves implementation of element erosion; as composites are brittle in nature and the formed chips in the machining process are dust like, element erosion has been commonly used in the literature for the modeling of composites machining (Isbilir & Ghassemieh, 2013; Santiuste et al., 2010). Hence, it was decided to model the orthogonal cutting of brass by using the element erosion approach in LS-Dyna instead. Material model MAT_015 Johnson-Cook in LS-Dyna has 70  been used for this purpose. The material flow stress formulation and input properties are as those shown in Equation ( 5-1) and Table  5-1. The main difference in the input parameters is the strain-to-failure which is used for the purpose of element erosion. The strain-to-failure in MAT_015 is calculated based on Equation ( 5-2) where    to    and       are input parameters.                                ̇                   (‎5-2) For simplicity however, all input parameters except for    is set zero. This means that the value of strain-to-failure at fracture is simply the value of   . Hence once the effective strain in an element reaches the input value, the element will be eroded from the mesh. An initial value of 300.0 % is assigned to this parameter. Different values for the strain-to-failure are tested however in order to better understand the sensitivity of the cutting force results to this particular parameter. 5.3.2 Model Description in LS-Dyna For the purpose of creating (meshing) the model in LS-Dyna, LS-PrePost has been used as the graphical interface. An orthogonal cutting model is set up where all the dimensions of both the cutting tool and the workpiece as well as the geometric features of the cutting tool such as rake and relief angles and the nose radius match that of the ABAQUS model also specified in Table  5-2. The two models are shown together in Figure  5.5. The model uses shell section Type 13 in LS-Dyna which is a two dimensional reduced integrated plane strain elements with unit thickness. Two dimensional automatic surface to surface contact algorithm available in LS-Dyna has been assigned to the rigid tool as the master surface and the deformable workpiece as the slave surface. Similar to the model developed in ABAQUS, the tool is considered rigid. The friction coefficient assigned to this contact is identical to that used in the ABAQUS model with a value of 0.15.. Similar to the ABAQUS model, the tool is assigned rigid constraints. Standard Hourglass Control Type 1 in LS-Dyna has been assigned to the model with default setting.   The boundary conditions in the LS-Dyna model are implemented slightly differently; instead of the cutting tool being fixed in space and the workpiece moving towards as in the ABAQUS, in LS-Dyna the workpiece is fixed and the rigid tool is assigned a velocity as previously shown in 71  Figure  3.1. The velocity assigned to the tool is identical to the velocity that is assigned to the workpiece in ABAQUS at 420 mm/s.   Figure ‎5.5_Comparison of the brass orthogonal cutting models in ABAQUS and LS-Dyna 5.3.3 LS-Dyna Results  Similar to the ABAQUS model, in the LS-Dyna model the cutting force, that is forces in the horizontal direction, are outputted and plotted versus displacement. The cutting force versus displacement for a strain-to-failure of 300.0 % (      ) is shown in Figure  5.6 along with the previously obtained ABAQUS results.  As the LS-Dyna model relies on element erosion based on the effective strain reaching a predefined value, a considerably higher noise level is expected to be present in the output. This is apparent from the plots provided in Figure  5.6. In order to enhance the results and reduce the noise level, damping in LS-Dyna was applied to the model with a system damping constant of 72  10% and 25% of the critical damping value for the first mode. For this purpose, first an eigenvalue analysis is conducted on the workpiece and the periods for the first three modes of vibration are extracted. The mass weighted critical damping constant value for the first mode of vibration is then calculated according to Equation ( 5-3) where     and    are the critical mass weighted damping value for mode   and the period of vibration of mode   respectively. In two separate models, 10% and 25% of the calculated damping value is assigned to the workpiece and through the Damping_Part_Mass in LS-Dyna; this is a mass weighted damping value represented by     in Equation ( 5-4).               (‎5-3)         ̈      ̇     (‎5-4) The damping constants assigned in LS-Dyna were obtained based on typical values previously used in finite element models. This was done only to check the sensitivity of the obtained results to the damping assigned to the system.   Figure ‎5.6_Cutting forces from orthogonal cutting of brass in LS-Dyna and ABAQUS 01234567890 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08Cutting Force (N/mm)Displacement (mm)ABAQUSLS-Dyna (D1=3.0, no Damp)LS-Dyna (D1=3.0, Damp=0.25(30875))LS-Dyna (D1=3.0, Damp=0.10(12350))73  A comparison of the cutting forces for the different LS-Dyna and ABAQUS models ran are shown in Figure  5.6. Despite the noise levels being higher in the LS-Dyna models due to the element erosion scheme, there is very good agreement between the cutting force obtained from LS-Dyna and ABAQUS on the orthogonal cutting of brass with the strain-to-failure set to 300.0 % in LS-Dyna.  Applying damping to the model shows minimal influence as the predicted cutting forces for the range of damping values used (no damping, and damping constants equal to 10 % and 25 % of the critical damping value for the first mode) are all within the same range with negligible difference.  5.3.4 Sensitivity Analysis of Results to the Strain-to-Failure Parameter  Before moving onto modeling of orthogonal cutting of composites, it was decided to check the sensitivity of the results to the strain-to-failure parameter. Therefore, the same orthogonal cutting model of brass set up in LS-Dyna is used and tested with different values of strain-to-failure,    as indicated in Table  5-3. No damping is assigned to these models.   Table ‎5-3_Strain-to-failures values used in LS-Dyna for the orthogonal cutting of brass Models Strain-to-failure    Model A 150.0 % Model B 300.0 %  Model C 450.0 % Model D 750.0 %  The results obtained from the three test runs is shown in Figure  5.7. As it can be seen from the plots, the cutting forces are highly sensitive to the strain-to-failure values up to 450.0 %. Beyond that, the strain-to-failure seems to have almost no effect on the results at all. This observation requires further investigation into the strain history of the individual elements undergoing plastic deformation.  74   Figure ‎5.7_Orthogonl cutting of brass in LS-Dyna for different strain-to-failure inputs (D1 = 150%, 300%, 450%, 750%) The elements that are deleted during the cutting process are outputted from the message file produced by the code and their strain histories are obtained and plotted. This data is extracted for the models with the strain-to-failure values    set at 450.0 % and 750.0 %. The effective plastic strain history for the first twenty four elements deleted in each case is obtained and plotted; the plots are shown in Figure  5.8 for the model with           and in Figure  5.9 for model with          . As indicated in these plots, most of the elements seem to fail well before reaching their assigned failure strain and with a higher strain-to-failure value, fewer elements tend to reach it prior to failure. In Figure  5.8, only one element seems to have reached an effective plastic strain of 750.0 % up to the displacement indicated and the rest fail at lower strain levels. In Figure  5.9 where    is 450.0 %, more elements reach the assigned values before being eroded from the model, however, most elements seem to undergo premature failure. This phenomenon explains why the cutting forces vary significantly when    changes from 150.0 % to 300.0 % and remain nearly unchanged when this parameter is varied from 450.0 % to 750.0 %; however, 0123456789100 0.02 0.04 0.06 0.08 0.1Cutting Force (N/mm)Displacement (mm)LS-Dyna (D1=1.5)LS-Dyna (D1=3.0)LS-Dyna (D1=4.5)LS-Dyna (D1=7.5)75  the time history of the individual elements must be examined more closely in order to investigate how the elements are being eroded.    By checking the individual elements showing this behaviour, it is found that the issue arises when elements undergo heavy distortion; hence, the element erosion feature is disabled and the model is reran. This allowed for recording the effective plastic strain values of some of the previously deleted elements and closer examination of the elements which exhibit such behaviour. It is found that in some elements where heavy distortion has occurred, the effective plastic strain suddenly ramps up in a very short time (less than the data recording frequency at every 0.000125   in the 0.25   run time) hence causing the element to reach very high strain values. It was observed that the effective strain jumps abruptly to 10’000.0 %. LS-Dyna support was contacted in this regard in order to seek for methods on avoiding such behaviour; unfortunately no solutions could be found.   Figure ‎5.8_Effective plastic strains for cutting model with D1 = 750.0 % shown for the elements deleted in the first 0.02 mm of cutting tool displacement  0.0%100.0%200.0%300.0%400.0%500.0%600.0%700.0%800.0%0 0.005 0.01 0.015 0.02Effective Plastic StrainTool Displacement (mm) Strain-to-failure of 750.0 %76   Figure ‎5.9_ Effective plastic strains for cutting model with D1 = 450.0 %  shown for the elements deleted in the first 0.02 mm of cutting tool displacement          0.0%100.0%200.0%300.0%400.0%500.0%0 0.005 0.01 0.015 0.02Effective Plastic StrainTool Displacement (mm) Strain-to-failure of 450.0 %77   Orthogonal Cutting of Composites: Numerical Simulation Chapter 66.1 Introduction In this study, the numerical simulations commenced with simulation of orthogonal cutting in metals and testing the element erosion scheme. With some confidence gained in the obtained results, the next step is to develop orthogonal cutting models in composites as it is the main objective of this study.  Most of the available literature on the numerical modeling of composites orthogonal cutting employ ABAQUS as the numerical platform. One such study by Santiuste et al. (2010) develops a numerical model using ABAQUS Explicit and performs sensitivity analysis on the various numerical parameters and tool geometries as described in Section Error! Reference source not found.. As a starting point for modeling orthogonal cutting of composites, it was decided to reproduce this model and obtain some preliminary results for comparison purposes. Next, the orthogonal cutting model is also developed in LS-Dyna and comments are made on the ability of such software to evaluate composite orthogonal cutting using the Equivalent Homogeneous Material approach (EHM). In this section, the model setup in ABAQUS and LS-Dyna are explained in detail.  6.2 Orthogonal Cutting of Composites in ABAQUS  As stated, with ABAQUS being the more common choice for the numerical code for composite machining purposes, orthogonal cutting model is first developed in ABAQUS Explicit. A two dimensional model is created in ABAQUS Version 6.14 where the dimensions used in the model for the different parts and geometric features are taken from Santiuste et al. (2010) and are summarized in Figure  6.1. The length is 3.0   and the height is 1.0   and the cutting depth is set at 0.2  ; the set cutting depth is very common in both numerical models and experimental investigations. 78   Figure ‎6.1_Workpiece dimensions in numerical modeling of orthogonal cutting of composites   The tool inset has rake angle α and relief angle γ of 5.0° and 6.0° respectively; similarly, the rake and relief angle values are very common in both numerical and experimental studies as listed in Table  2-1. The geometrical features of the tool are summarized in Figure  6.2. The cutting edge radius of the tool is initially set to 50    based on the model developed by Santiuste et al. (2010) which replicates the experimental investigation by Bhatnagar et al. (2004) The cutting edge radius however was then changed to 5    as discussed in Section  Chapter 7. (Bhatnagar, Nayak, Singh, Chouhan, & Mahajan, 2004; Soldani et al., 2011).     Length = 3.0 mmHeight = 1.0 mmCutting depth = 0.2 mm79   Figure ‎6.2_Tool insert dimensions in numerical modeling of orthogonal cutting of composites 6.2.1 Mesh Size and Element Type For the purpose of meshing the workpiece, a mesh size of 5    is suggested by Santiuste et al.; this however yields an extremely dense mesh which is not very appropriate and efficient for the purpose of the numerical run. Therefore, the workpiece is partitioned into different zones in ABAQUS as shown in Figure  6.3 a). The region located at the cutting plane where the tool comes in contact with the workpiece is uniformly meshed with 4 noded quad elements with size of 5    by 5   ; as the elements in this region are expected to undergo very large strains and deformations, it is necessary to ensure the mesh is refined and the elements have proper aspect ratios. In order to achieve the desired mesh, a large number of partitioned faces and meshing schemes were used to arrive at the desired mesh presented in Figure  6.3 b). During the initial trial runs, it was found that a few elements located on the curved region at the corner where the tool initially comes in contact with the workpiece results in numerical issues due to excessive distortion; these elements were identified and removed or combined with neighbouring elements in order to adjust the aspect ratios. These issues arising during the numerical runs will be more thoroughly discussed. R.PLength = 1.0 mmHeight = 2.0 mm80   Figure ‎6.3_Workpiece setup in ABAQUS for orthogonal cutting of composites; a) partitioning of the workpiece for different mesh densities, b) meshed workpiece  The model employs reduced integrated plane stress elements denoted as CPS4R with linear formulation available in ABAQUS Explicit. Hourglass Control has been enabled using the stiffness type damping. As previously explained in Sections  Chapter 2 and Error! Reference source not found., orthogonal cutting of composites is considered to be a plane stress phenomena due the excessive out of plane displacement observed during experimental studies of orthogonal cutting (Nayak et al., 2005,b; Wang et al., 1995,a; Arola et al., 2002). Uniform 5 by 5 μma)b)81  For the purpose of meshing the cutting tool, a similar approach has been used whereby the region in the vicinity of the cutting edge where the tool is in contact with the workpiece and the formed chip contains a very refined mesh in the same order as the mesh on the workpiece. Even though the cutting tool is assigned a rigid constraint, the mesh at the contact needs to be as refined as the deformable part for accurate modeling of contact algorithm and for avoiding penetration of the two parts.  6.2.2 Hashin’s‎Damage‎Model:‎Input‎Parameters  Hashin’s constitutive model available in ABAQUS with progressive damage (Section  4.2) has been assigned to the workpiece. The elastic properties of the composite are as listed in Table  4-1 which are taken the numerical study from by Santiuste et al (2010).  For the damage properties of the composite which include the fracture energy densities for longitudinal and transverse tension and compression loading, initially the values used by Santiuste et al. (2010) listed in Table  6-1 were used as inputs. The fracture energy values assigned to the longitudinal failure modes are 3 orders of magnitude less than the typical fracture energy values used for damage modeling of CFRP composites. It must be noted that the input value for the fracture energies is purely meant to provide some orders magnitude for toughness and energy dissipation in the system. Santiuste et al. (2010) used the fracture energy input to distinguish between brittle and ductile composites. For glass fibres, higher values of fracture energy are used to reflect the more ductile nature of GFRPs compared to CFRPs. As previously mentioned, many numerical models delete the elements in a brittle (instantaneous) manner based on some strength criteria without any evolution of damage (Isbilir & Ghassemieh, 2013; Nayak et al., 2005,b; Santiuste et al., 2010).  The reason for not using the established values of fracture energy for the longitudinal direction is the nature of the orthogonal cutting itself. The depth of cut is in the order of 0.2 mm and the cutting process involves cutting and shearing of individual fibres and fracturing of the fibre matrix interface as discussed in Chapters 2 and 3. Hence the cutting process involves fracturing of the individual fibre and the matrix phases of the composite rather than fracturing of the composite as a unit. Previously values of 100’000       (100      ) has been used in numerical models developed at the UBC for the purpose of composite tube crushing. The Over-height Compact Tension (OCT) experiments from which the fracture energy values are computed and 82  obtained are also conducted on composite specimens with sizes in the order of few centimeters where the crack and the damage height reaches 20 mm which are an order magnitude greater than the damage zone and cutting depths in orthogonal cutting (McGregor, Vaziri, Poursartip, & Xiao, 2007; Zobeiry, Forghani, McGregor, Vaziri, & Poursartip, 2008). As a result, it is believed that due to the scale at which cutting takes place, the properties of the individual phases is more relevant than the damage properties of the composite unit. Therefore, it is considered inappropriate to assign the typical fracture energy values to orthogonal cutting where the cutting depth is 0.2  .  Table ‎6-1_Damage parameter (fracture energy density) inputs for Hashin's damage model in ABAQUS applied to CFRP composites used by Santiuste et al (2010)  Longitudinal (Fibre) Tension 160       Longitudinal (Fibre) Compression 200      Transverse (Matrix) Tension 300      Transverse (Matrix) Compression 600      6.2.3 Boundary Conditions and Contact in ABAQUS The boundary conditions applied onto the composite workpiece are as shown in Figure  3.1 whereby the base of the workpiece is fixed in the two in plane translational directions and the vertical sides are only allowed to move in the Y-direction and are restrained from moving horizontally.  The cutting tool, as previously mentioned, is assigned a rigid constraint. In ABAQUS, a rigid part is assigned a reference point to which the boundary conditions are applied. In the cutting model, the reference point is chosen to be the point indicated in Figure  6.2 by “R.P”. The reference point is fixed in the vertical direction and assigned an initial velocity in the horizontal direction (towards the workpiece). Initially, a velocity of 0.0833      (5.0     ) is assigned to the reference point in the negative X direction for 0.25   for a total tool displacement of 2.0  . This value is considerably greater (up to 10 times) than the cutting speeds in most experimental studies as summarized in Table  2-1. However, it was found necessary to increase the cutting speed in order to reduce the analysis time as the time steps are extremely small (in the order of      to       ) and therefore requires many steps to 83  complete. Hence increasing the cutting speed was found to be an efficient and effective way of reducing the run times. The contact between the two parts has been defined through a surface-to-node interaction where the rake surface, relief surface (Figure  1.1) and the edge radius has been defined as the master surface and the nodes within the cutting plane are defined as the slave surface. The reason that the slave surface has been defined as the set of nodes within the 0.2   cutting depth rather than the exterior surface of the workpiece is due to the element erosion scheme. As elements erode, new surfaces are created for which contact is not defined, therefore a set of nodes within the interior of the workpiece where contact could be possible is defined under the slave surface. The contact property is assigned by defining a tangential behaviour with a friction coefficient of 0.5 which is a commonly used value; the friction formulation is enforced using the penalty method.  As the material model is orthotropic, local material orientation with respect to the global X-Y axes must be defined where the 1-direction denotes the longitudinal (fibre) direction and 2-direction denotes the transverse direction. The angle of each fibre orientation is measured in a clockwise manner from the horizontal X axis as shown in Figure  6.4.  Figure ‎6.4_Material orientation angles in ABAQUS 6.3 Orthogonal Cutting of Composites in LS-Dyna The orthogonal cutting model is also developed in LS-Dyna for the purpose of this study as the CRN has much experience and expertise with LS-Dyna and as this numerical platform is well known for its explicit capabilities.  120 degrees45 degrees90 degrees12XY84  The dimensions used in creating the different components of the model are identical to that used in the ABAQUS model explained in the previous section. The geometric features of the cutting tool are identical as well with the cutting edge diameter set to 5    (0.005  ).   6.3.1 Mesh Size and Element Type  Similar to the model developed in ABAQUS, the smallest element size located with the 0.2   cutting depth region of the workpiece is 5    by 5   . The meshing capabilities of LS-PrePost,the graphical interface used for LS-Dyna, are much more limited than ABAQUS; hence meshing of the model to obtain the desired mesh with a high density of elements where significant deformation is expected, and coarse elements elsewhere, was obtained through many cycles of trial and error.  For the cutting tool, the mesh density is refined at the cutting edge and its vicinity, and coarser further away from the vicinity of contact with the workpiece. For comparison purposes, the meshed models from the two software are provided in Figure  6.5.  The model is assigned shell section with element formulation, Type 2_Belytschko-Tsay, which is a plane stress formulation and the default element in LS-Dyna. Default hourglass control was initially assigned to the model, but then changed to hourglass Type-4 which is a stiffness type hourglass control due to excessive hourglassing of the elements that resulted from using the default hourglass control; as a result, the hourglass control in ABAQUS was also changed from the default setting to stiffness type.  85   Figure ‎6.5_Meshed models in ABAQUS and LS-Dyna 6.3.2 MAT_054 Damage Model: Input Parameters  The material model used for damage modeling of composites in LS-Dyna has been described in detail in Section  4.3. The elastic properties of the CFRP are identical to those used in the ABAQUS model listed in Table  4-1. The only difference in the elastic properties is that MAT_054 requires the Poisson’s ratio to be input as     rather than the more common notation of    .  As previously explained, the damage parameters define the strain-to-failure for various modes of failure. These strain-to-failure values are used as input for the tension and compression in both longitudinal and transverse directions. No standard value exists for this parameter, therefore multiple combination of values were tested which is explained in the Results Section.  As the material model is orthotropic, local material orientation has to be assigned to the model in order to simulate orthogonal cutting of the desired fibre orientation. MAT_054 in LS-Dyna ABAQUSLS-Dyna5 by 5 μm element size5 by 5 μm element size86  allows this to be done using several different ways. The method used in the orthogonal cutting simulations in this study is through defining the primary 1 and 2 material axes by vectors relative to the global coordinate system as shown in Figure  6.6.   Figure ‎6.6_Vectors defining material orientation angles in LS-Dyna 6.3.3 Boundary Condition and Contact in LS-Dyna The boundary conditions imposed on the workpiece and the cutting tool are identical to those used in ABAQUS whereby the workpiece is fixed and the tool is assigned a velocity towards the workpiece.   For defining the contact between the two components, 2D_Automatic_Surface_To_Surface contact available in LS-Dyna for two-dimensional models is used. The rigid tool is assigned to the master surface and the workpiece is assigned to the slave surface. Similarly, friction coefficient of 0.50 is used between the surfaces.       0 degrees1=(1,0,0)2=(0,1,0)XY 45 degrees90 degrees2=(1,0,0)1=(0,1,0)87   Results and Discussion  Chapter 7The objective of this study is to obtain cutting forces from the numerical models and evaluate the ability of the finite element method for studying machining of composites. This area of study is relatively new and under-developed especially in the area of numerical modeling. A thorough background and review of the studies available on the numerical modeling of composites has been provided in Section  3.1 and the shortcomings and areas where more research and study is required to further develop this field is provided in Section  2.7 and  3.5.  In this section, cutting forces obtained from the numerical models developed in ABAQUS and LS-Dyna is presented and discussed. Cutting forces are obtained for orthogonal cutting of uni-directional CFRP with fibre orientations of 0°, 45°,90° using an equivalent homogeneous material with orthotropic material properties. The results of the numerical runs and issues encountered in the simulations are discussed.  7.1 Cutting Force Plots From ABAQUS The details of the numerical model along with the elastic properties of the material used to represent the orthotropic CFRP composite is provided in Section  6.2. The initial set of models ran in ABAQUS Explicit employs Hashin’s damage input parameters as used by Santiuste et al. (2010) in Table  6-1 and repeated here (Table  7-1) for convenience. The series of fibre orientations ran with these damage properties is referred to as the “benchmark” model identified in Table  7-1. The terms FT, FC, MT and MC refer to fibre tension, longitudinal (fibre) tension and compression and transverse (matrix) tension and compression respectively and “Max Degrdation is as defined in Section  4.2.3        88   Table ‎7-1_"Benchmark" model description in ABAQUS Model Identification Cutting Speed Edge radius FT FC MT MC Max Degradation      benchmark_R5micm 5.0      (0.0833      ) 5    160 200 300 600 1.0 benchmarkh_R5micm_fast 50.0      (0.833      ) 5    160 200 300 600 1.0 benchmarkh_R5micm_fast_MD0.98 50.0      (0.833      ) 5    160 200 300 600 0.98  The cutting forces obtained from the “benchmark” series are plotted for 0°, 45° and 90° fibre orientation in Figure  7.1 a) through c). For each fibre orientation, the cutting force plot is obtained for two different cutting speeds of 5.0 and 50.0      in order to see how the results vary. The cutting forces are plotted up to the displacement at which the computation failed to proceed any further due to presence of excessively distorted elements. The black line in the plots represents the 25 point moving average in order to attenuate the noise in the raw data and show the steady state cutting forces obtained.  89    Figure ‎7.1_ Cutting forces versus displacement for a) 0°, b) 45° and c) 90° fibre orientations for cutting speeds of 5.0 (green line) and 50.0 (red line) m/min 0102030405060700 0.1 0.2 0.3 0.4 0.5 0.6Cutting force (N/mm)Displacement (mm)Benchmark_R5micm_fast Benchmark_R5micm_NormalSpeed01020304050600 0.02 0.04 0.06 0.08 0.1 0.12 0.14Cutting force (N/mm)Displacement (mm)Benchmar_5micm_fast Benchmark_R5micm_NormalSpeed01020304050600 0.1 0.2 0.3 0.4 0.5 0.6Cutting force (N/mm)Displacement (mm)Benchmark_R5micm_Fast Benchmark_R5micm_NormalSpeeda)b)c)90  As shown, a significant amount of noise is present in the raw data; this is mainly due to the brittle failure of the element. For all fibre orientations and cutting speeds, the data is plotted for equal displacement intervals of           . As mentioned, depending on the energy level inputs, the ductility of the elements can be adjusted. With the values used in the “benchmark” models (Table  7-1), a very small amount of displacement in the longitudinal direction is required in order to reach complete damage saturation of the 5    by 5    elements. The required displacement for reaching the saturation for each mode is obtained by the expression shown in Figure  4.1. For the longitudinal tension and compression, the saturation displacements are             and             respectively. These values are considerably lower than the data recording frequencies and therefore the progression of damage in the models is not captured and elements fail in a brittle manner and yield the force plots shown.  The steady state cutting forces obtained from the numerical simulations are summarized in Table  7-12. The cutting force is presented for the first 0.05   of tool displacement and for the entire duration of the simulation run. The reason for this is that in the study by Santisute et al. (2010), only the initial portion of the run has been investigated and presented. Figure  7.6 shows the cutting forces obtained by Santiuste et al. (2010) for 45° fibre orientation in the first 0.08   of tool displacement (cutting speed of 0.5     ). In the cutting force plots obtained (Figure  7.1), the plots change characteristic past a certain point; this change is apparent from a sudden jump in the cutting forces (0° and 45° at just under 0.1   tool displacement) or a change in the noise level present (90° fibre orientation just under 0.07   too displacement).   The value of the cutting forces obtained for the 0° and 45° fibre orientations are within the range of values obtained from experimental results as indicated in Figure  2.8 and Figure  2.12; for 90° fibre orientation however, a steep increase in the cutting forces in shown in most experimental studies compared to the forces in 0° and 45° fibre orientations which is not captured in the numerical models.     91  Table ‎7-2_Cutting forces per unit width of UD CFRP obtained from the "benchmark" model (cutting depth of 0.2 mm) Fibre orientation (degrees) Average cutting force (    ) First 0.05 mm Entire duration  0 20 43 45 34 34 90 37 37  As indicated by the cutting force plots, the model with the higher cutting speed tends to run much further than the model with the lower cutting speed; the cutting speed however has almost no influence on the cutting force plots and the two plots for each fibre orientation are nearly identical. Hence reducing the cutting speed while maintaining all other parameters of the numerical model constant causes the solver to encounter an excessively distorted element much earlier through the analysis. Investigating this phenomenon further, it is believed that the reason for such behaviour lies in the material model. As explained in Section  4.2, Hashin’s damage model requires the damage parameter for the longitudinal direction to reach a value of 1.0 in order to erode the element. As indicated in Figure  4.4, initially the damage parameter increases very rapidly and as it approaches the point of saturation, it flattens and damage accumulates at a lower rate. In the orthogonal cutting model, as elements undergo damage and their elastic properties are degraded, certain elements reach damage parameter values that are just below unity (    ) and hence not yet removed from the mesh. However, due to having lost nearly all their elastic properties, these elements can undergo significant deformation. Also, some elements undergo damage in the transverse direction and their properties are degraded in this direction. However, the code only erodes the element if saturation is reached in the longitudinal direction. In the orthogonal cutting model, the elements undergo multidirectional loading as opposed to the ideal cases discussed in Section  4.2.4 whereby single elements undergo uniaxial loading; hence an element can undergo significant deformation due to lack of resistance in the transverse direction while having load carrying capacity in the longitudinal direction. It is believed that the lower cutting speed causes the elements to be trapped in a very fine region where damage has not saturated fully, but properties have degraded enough to allow excessive deformation of the elements. The stable time step in the finite element model which is governed by the elements 92  size, bulk modulus and density of material is constant regardless of the cutting speed; therefore a lower cutting speed allows for smaller increments in the element history to be captured and increases the chances of the element becoming trapped between the nearly full damage and full degradation. In order to gain further confidence into this reasoning, another set of models were also ran with the cutting speed reduced to 1/10th of the “Normal Speed”. In this case the models encountered excessively distorted elements much earlier than indicated by the green lines in Figure  7.1. In order to overcome this issue, it was decided to make use of the “Max Degradation” parameter which is part of the element erosion control. This parameter defines the damage level at which the element properties are set to zero and the element is eroded. As explained, with progression of damage, the rate at which damage is accumulated reduces and this in turn causes some elements to be trapped in a fine zone. The default value for Max Degradation is 1.0; however, by inputting a lower value for this parameter, we allow the properties of the damaged elements to be fully degraded and the element removed while there is some residual stiffness and load carrying capacity in the element. The same set of models are therefore rerun with the damage parameter set at 0.98 (Table  7-1). In the numerical study of composite orthogonal cutting conducted by Lasri et al. (2009) where stiffness degradation method is implemented, once damage has developed, the stiffness is reduced to 0.05 of the initial value in order to avoid numerical problems (Lasri, Nouari, & Mansori, 2009) The results from the models with the reduced and default value of Max Degradation are plotted in Figure  7.2. The cutting forces are nearly identical up to a certain cutting tool displacement, but then increase; although this is in contrary to our expectations, the results obtained from ABAQUS are deemed unreliable past a certain point due to the presence of too many heavily distorted elements which in turn yield inaccurate results. Hence, the plots must be correlated with the point of complete chip formation in the model and the cutting forces should be taken at those points. Past the initial chip formation point which is caused by deletion of a few elements, many damaged elements which are not eroded undergo heavy distortion and the models are deemed too unreliable once they reach that phase.  93   Figure ‎7.2_ Cutting forces versus displacement for a) 0°, b) 45° and c) 90° fibre orientations for model with Max Degradation set to 1.0 (green line) and set to 0.98 (red line) 0204060801001200 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4Cutting force (N/mm)Displacement (mm)Benchmark_R5micm_fast_MD0.98Benchmark_R5micm_fast0204060801000 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4Cutting force (N/mm)Displacement (mm)Benchmark_R5micm_fast_MD0.98Benchmark_R5micm_fast0204060801000 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4Cutting force (N/mm)Displacement (mm)Benchmark_R5micm_fast_MD0.98Benchmark_R5micm_fasta)b)c)94  The noise level present in the simulation is reduced when the Max Degradation is reduced to 0.98 and the run times are also reduced. In the case of 90° fibre orientation, the model runs much further than the previous case with the default Max Degradation value of unity. The “Displacement” axes on the three plots in Figure  7.2 are plotted up to 0.4 mm of displacement for purpose of comparison; as shown, in all three fibre orientations, the cutting forces increases up to a displacement of 0.02   as the angled rake surface of the tool comes into full contact with the workpiece. Afterwards, the forces increase steadily but at a lower rate; in order to capture the portion of the plot which represents the chip initiation, the numerical simulations are checked. For each fibre orientation, the time at which a chip is deemed to have initiated is recorded and the cutting forces are read from the plots at that instant. In Figure  7.3 a) through d), multiple stages of chip formation in the case of 0° fibre orientation is shown. Based on the experimental observations, the chip in this orientation is formed through formation of cracks along the cutting plane and bending of the chip. In Figure  7.3 d) the extension of matrix damage along the cutting plane is shown which indicates the chip formation. This occurs at roughly 0.07   tool displacement which based on the results in Figure  7.2 a), yields a cutting force value of 25-30     .      In the case of 45° and 90° fibre orientations, the chip formation occurs through shearing in the matrix and matrix cracking followed by fibre fracture. From the numerical models, the instant of chip formation for 45° 90° fibre orientation is shown in Figure  7.4 a) and b) respectively; for the 45° fibre, the shear damage is shown while for the 90° fibre orientation, the matrix tensile damage has been shown. The chip formations occur at displacements of 0.076 and 0.25   for the 45° and 90° orientations respectively which according to the force plots in Figure  7.2 b) and c), yield a cutting force of 38-40 and 78-80      respectively. It must be noted that for the 90° orientation, a considerably larger tool displacement is required to reach complete chip formation as the material is orientated in the soft (transverse) direction with respect to the tool displacement. The cutting force results from the models with reduced value for Max Degradation are summarized in Table  7-3. The model is extended to fibre orientations of 30°, 60° and 135° as well and the cutting force at the instant of chip formation is recorded and stated in Table  7-3.  95   Figure ‎7.3_Chip formation in 0° fibre orientation with Max Degradation set to 0.98 (matrix tensile damage shown)  Figure ‎7.4_Chip formation in a) 45° and b) 90° fibre orientations   b)a)d)c)Complete chip formation; extension of matrix tension damage along the cutting planea) 45 degree b) 90 degree96  Table ‎7-3_Average cutting force values from ABAQUS model at complete chip formation with Max Degradation set to 0.98 Fibre orientation (degrees) Average cutting force with cutting depth of 0.2 mm (    ) at instant of complete chip formation  0 20-23 30 42-44 45 38-40 60 55-58 90 78-80 135 65-67  The cutting forces presented in Table  7-3 indicate a large jump in the cutting forces for 90° fibre orientation which is observed by most experimental studies. The forces obtained are within the same range as the experimental results on CFRP orthogonal cutting with similar cutting depths as shown in Figure  2.7, Figure  2.8 and Figure  2.13. The models are therefore capable of capturing the global response reasonably well; however, improved damage models and damage properties appropriate for this purpose are required in order to further enhance the simulation capabilities. A comparison of the obtained cutting forces with some experimental data on CFRP orthogonal cutting is provided in Figure  7.5. As mentioned, the numerically obtained forces fit well with the experimental results given the experiments are each conducted on materials with slightly different properties with different test specifications. The experimental details are provided in Table  2-1 (Bhatnagar et al., 1995; Wang et al., 1995,a).   97   Figure ‎7.5_Comparison of experimentally obtained cutting forces for multiple fibre orientations in CFRP orthogonal cutting The models tend to suffer from element distortions. During the analysis, intrusion of the elements within the workpiece as the initial chip is formed is observed. This is due to the fact that contact between the elements within the workpiece is not defined and therefore as some elements erode and form a chip, the formed chip intrudes the workpiece at which point the model is considered to be invalid. Theoretically, the distortion control feature which is enabled should cope with distorted elements and remove them from the mesh; however, many distorted elements remain. The results presented by Santiuste et al. (2011) also focus only at the initial 0.1   of displacement. The cutting force result is only presented for 45° fibre orientation as shown in Figure  7.6 which indicates good agreement with what has been obtained for 45° fibre orientation. The results obtained for 45° fibre orientation are overlapped on the plot for comparison. Due to difference in the edge radius, the ramp to the steady state varies in the plots. The cutting force history plots are not presented for other fibre orientations in the study by Santiuste et al. and other studies of similar nature (Soldani et al., 2011).   0204060801001201401601802000 15 30 45 60 75 90Force per unit width (N/mm)Fibre orietnation (degree)ABAQUS (current study)Cut Depth = 0.2 mm, Venu Gopala Rao et al. (2007)Cut Depth = 0.25 mm Bhatnagar et al (1995)Cut Depth = 0.254 mm, Wang et al.(1995)"98   Figure ‎7.6_Cutting force versus time for 45° UD-CFRP cutting for multiple mesh types in ABAQUS (taken from Soldani et al., 2011)  7.2 Cutting Force Plots From LS-Dyna The cutting model is also created in LS-Dyna. The details of this numerical model is provided in Section  6.3 whereby all the details in the ABAQUS model has been recreated. The material model employed to represent the CFRP composite is MAT_054 which is explained in Section  4.3. The elastic properties of the material are as listed in Table  4-1. The damage parameters in this material model are the strain-to-failure values. Two sets of input values are used and referred to as “Set 1” and “Set 2”; the difference between the two sets of models lies in the strain to failure input for the longitudinal (fibre) tension and compression whereby the strain-to-failure values in the longitudinal mode in “Set 2” is three times those used in “Set 1”. The values of the strains in “Set 1” are calculated such that the energy dissipated at the point of fracture and erosion matches the values that were previously used in the ABAQUS models as specified in Table  7-1. The values for the input parameters are summarized in Table  7-4. The motivation for doing so is to see the variation in the cutting force results for each fibre Benchmark_R5micm_fast_MD0.9899  orientation and the dependency of the forces on the damage parameters. There are no standard or established values for these parameters and more specifically for the purpose of orthogonal cutting. It must be noted that the input values are meant to provide some level of ductility and energy absorption capacity.  Table ‎7-4_LS-Dyna strain-to-failure values used in MAT_054 Model Strain-to-failure Values  FT FC M Set 1 0.075 0.100 1.0 Set 2 0.024 0.0334 1.0  The cutting forces are plotted in Figure  7.7 a) through c) for fibre orientations of 0°, 45° and 90°. The data is plotted at every            intervals of displacement.  The simulations in the LS-Dyna are much less noisy and without the issues involving distorted elements as was the case in ABAQUS simulations. Despite using a different numerical code, a number of reasons are believed to be contributing to this observation. The material model (MAT_054) used in LS-Dyna has an elastic perfectly plastic behaviour in which past the yield point under uniaxial load, the material plateaus at its yield strength up to the strain-to-failure defined. Since there is no degradation of the material properties, as an element yields in any mode, the stress on the surrounding elements can be maintained without unloading and hence there is no localization of damage to a series of elements. Secondly, as the model allows for erosion of elements in the transverse direction in addition to the longitudinal direction, erosion takes place much more frequently; although this allows the runs to go further without creating too many distorted elements, the downside is that fracturing in the longitudinal direction is not required to erode the element. This however is an essential part of the chip formation as previously explained where formation of chips involves fracturing of the fibres.   100   Figure ‎7.7_Cutting forces versus displacement for 0°, b) 45° and c) 90° fibre orientations for models with different value of strain-to-failure  01020304050600 0.2 0.4 0.6 0.8 1 1.2 1.4Cutting force (N/mm)Displacement (mm)Set 1 Set 2051015202530350 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Cutting force (N/mm)Displacement (mm)Set 1 Set 20510152025303540450 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Cutting force (N/mm)Displacement (mm)Set 1 Set 2a)b)c)101  In the case of 0° fibre orientation for “Set 1” and the 45° fibre orientations, the drops in the forces coincide with the instant of chip release in the model. The model is able to capture the entire chip formation process and the subsequent chip release as opposed to the models in ABAQUS whereby only the chip initiation is captured. The chip formation and chip release for 0° and 45° fibre orientations runs is as indicated in Figure  7.8 a) and b) respectively. In the 0° case, the dilation of chip in a mode I form followed by compression failure in the longitudinal direction leads to breakage of the chip. In the case of 45° fibres, the chip is released along the fibre-matrix interface oriented at 45°. This is very much in line with the experimental observations based on multiple studies (Wang et al., 1995,a; Zitoune et al., 2005).   Figure ‎7.8_Chip formation stages in a) 0° and b) 45° fibre orientations As indicated by the cutting force plots, the runs proceed further and with much more consistency compared to the ABAQUS runs. The reason as mentioned is believed to be due the difference in the element erosion scheme and lack of damage localization due to the difference in the material model. The cutting forces obtained show great dependency on the damage parameters in the longitudinal direction for the case of 0° fibre orientation. For the 45° and 90° fibre orientation, the longitudinal damage parameters have no influence on the force history plots which indicates that the cutting forces and chip formation is entirely dependent on the transverse (matrix) properties.  Although the chip formation process is well predicted, the cutting force values and trends obtained from LS-Dyna tend to vary from the experimentally obtained values. This is especially true for the 90° fibre orientation where the deletion of the element is due to failure in the b)a)102  transverse direction with no longitudinal failure involved. The force history plot also indicates that the results are independent of changes with changes in damage properties in the longitudinal direction.                     103   Conclusions and Future Work  Chapter 8In this work, a thorough study of the existing literature on the orthogonal cutting of composites has been conducted which covers experimental and numerical work conducted in this field and presents the findings up to this point in time. As part of this research, a numerical study of CFRP orthogonal cutting is conducted as well using ABAQUS and LS-Dyna as the numerical platforms. By using an equivalent homogeneous orthotropic material to represent the CFRP and creating a numerical model in the two-dimensional space, reasonable estimates of the cutting forces of uni-directional composite cutting was achieved. Although the value of cutting forces obtained for the tested fibre orientation were within the range of values obtained from experimental studies of similar type, better understanding of damage behaviour of the material especially at the micrometer scale in which cutting takes place is needed. Depending on what type of damage model is being used to simulate the damage response of the material, the results tend to be sensitive to certain damage parameters and therefore better representation of the material damaged properties is necessary in order to gain confidence in the results and be able to apply the same approach to a broader range of materials and applications.  Based on the literature review conducted, it is found that despite many attempts to use the orthogonal cutting approach as a way to better understand the damage mechanism and material removal in more complicated machining processes such as drilling, no study has been found where orthogonal cutting of composites is transformed into a drilling or milling operation analytically and where the results have been compared with drilling or milling experiments conducted on the same material. This is believed to be a major missing gap in the existing knowledge and it is considered of great value to take the next step and link the orthogonal cutting to a more complicated operation such as drilling through geometric transformations and confirm the validity of this approach with experiments on the same material.    104  Bibliography Abena, A., Leung Soo, S., & Essa, K. (2015). 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