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Numerical Methods Applied Id Metallurgical Processing

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Numerical Methods Applied Id Metallurgical Processing NUMERICAL METHODS APPLIED ID METALLURGICAL PROCESSING by JUAN HECTOR BIANCHI A thesis submitted for the degree of Doctor of Philosophy of the University of London and for the Diploma of Membership of the Imperial College John Percy Research Grou Department of Metallurgy Royal School of Mines, Imper i a 1 Co 11ege London JULY 1983 In bulk forming operations the plastic deformation is very large compared with the elastic one. This fact allows the use of constitutive laws based upon both current stress-strain rate measures. A Finite Element formulation of the mechanics involved in quasi-steady state conditions was implemented, introducing the incompressibi1ity via a penalty on the volumetric strain rate. The perfectly plastic behaviour was considered first, and the extrusion process was thoroughly analyzed. Solutions for direct and and indirect modes of operation up to very high extrusion ratios are presented. Both plane strain and axisymmetric geometries were considered. Frictional conditions at the billet-container interface were introduced in two ways. Comparison between results obtained using these lines of approach and numerical problems associated to them are examined. Pressure results are compared with previously reported solutions resulting from Slip Lines, Upper Bounds and Finite Elements. The behaviour of the internal mechanics as a function of changes of geometrical and frictional conditions are examined. Next, a non-linear flow stress resulting from hot torsion experi- ments was considered. The thermal analysis involves the solution of another equation, which is coupled with the mechanical problem due to convection, heat generation and flow stress dependence on temperature. After an appropriate transformation, that equation is formulated in a "weak" form and implemented to be solved jointly with the mechanical part. Application was made to quasi-steady-state analysis of axisymmetric extrusion. The dependence of solutions with billet length, tool parameters and numerical instabilities are examined and comparison is made with experimental results. Finally, a review of dynamic analysis research on forming problems is carried out. The difficulties involved in implementing such formulations from the state of the work of the present thesis, and their consequences in the transient situation detected experimentally in extrusion processes are discussed. TABLE OF CONTENTS Page No ABSTRACT i i i LIST OF FIGURES Vi i LIST OF TABLES x NOMENCLATURE xi 1. INTRODUCTION AND LITERATURE SURVEY 1.1 Introduction 1.2 Analysis of Metal Working Processes 1.2.1 Experimental, empirical and hybrid methods 1.2.2 Slip line field techniques 1.2.3 Upper bound solutions 1.2 .4 Fi n i te elements 1.2.4.1 Elastoplastici ty 1.2.4.2 Viscoplasticity 1.3 Summary 22 2. BASIC CONCEPTS AND FINITE ELEMENT FORMULATION 2.1 I ntroduct ion 26 2.2 Mechanical Problem 26 2.2.1 Strain 2.2.2 Stress 2.2.3 Constitutive models 2.2.3.1 Elastoplasticity 2.2.3.2 Rigid-Plastic 2.2.3.3 Viscoplasticity 2.2.4 Variational principle - viscoplasticity model with penalty formulation 2.2.5 Finite element treatment 2.2.6 Stress analysis 2.3 Thermal Problem 43 2.3.1 Basic equations 2.3.2 Finite element treatment 2.4 47 Thermomechanic Coupling 49 2.5 Solution Procedure 3. EXPERIMENTAL 3.1 I ntroduct ion 54 3.2 The Extrusion Press 54 3.2.1 Container heating 3.2.2 Bi1 let preheat 3.2.3 Extrusion data recording 3.2.4 Direct extrusion tooling 3.2.5 Indirect extrusion tooling 3.2.6 Water quench 3.3 Experimental Procedure 61 3-3-1 Direct extrusion 3.3.2 Indirect extrusion 3.4 Materials 63 1 v 4. ANALYSIS OF EXTRUSION I: PERFECTLY PLASTIC IDEALIZATION 4.1 Introduction 65 4.2 Direct Plane Strain Extrusion Through Square Dies 68 4.2.1 Computational conditions 4.2.2 Direct extrusion, fc=2 4.2.2.1 Smooth container - smooth die 4.2.2.2 Rough container - rough die 4.2.2.3 Direct extrusion, R=10 4.3 Indirect Plane Strain Extrusion Through Square Dies 83 4.3.1 Computational conditions 4.3.2 Indirect extrusion, FL=2 and FL= 10 4.4 Alternative Introduction of Friction 92 4.4.1 Computational conditions 4.4.2 Mean strain rate 4.4.3 Plane strain extrusion 4.4.4 Axisymmetric rod extrusion 4.4.4.1 Direct extrusion, low ratios 4.4.4.2 Direct and indirect extrusion, high rat ios 4.5 Conclusions 113 5. ANALYSIS OF EXTRUSION II: THERMOMECHANICAL WORKING 5.1 Introduction 117 5.2 Mechanics H7 5.2.1 Material behaviour 5.2.2 Acceleration terms 5.2.3 Applicability of the steady state approach. I: assumptions and general considerations 5.3 One Dimensional Thermal Model 127 5.4 Extrusion Thermal Model 1 ing 133 5.4.1 General considerations 5.4.2 Applicability of the thermal model 5.5 Direct Rod Extrusion R=12.4, A15456 136 5.6 Direct Rod Extrusion R=20, Al5052 152 5.7 Higher Extrusion Ratios 156 5.8 Applicability of the Steady State II: Transient Limit and Steady State Flows 170 5.9 Conclusions and Recommendations for Further Work (l) 178 5.9.1 Post-peak load steady state 5.9.2 Initial transient 6. DYNAMIC ANALYSIS OF THE EXTRUSION PROCESS 185 6.1 Introduction 6.2 El as topiastic Model 185 6.2.1 General considerations 6.2.2 Variational principle 6.2.3 Frame of reference 6.2.4 Constitutive relationship 6.2.5 Integration of the equations of motion v 6.3 Viscoplastic Model 6.4 Conclusions and Recommendations for Further Work (II Appendix I: COMPUTATIONAL FORMS FOR FINITE ELEMENT ANALYSIS Appendix II: DATA ADO.UISITION AND GENERATION ACKNOWLEDGEMENTS REFERENCES LIST OF FIGURES Page No Chapter One Fig 1.1 Isothermal Dynamic Analysis. Qualitative behaviour of load-displacement FE reported results 2b Chapter Two Fig 2.1 Frictionless interface b2 Fig 2.2 Sticking interface b2 Chapter Three Plate I : General layout of the extrusion press 55 Plate I I a) Di rect tooli ng 60 b) Ind i rect tooli ng Chapter Four Fig b. 1 FE mesh for extrusion through square dies 69 Fig 4.2 Direct Extrusion - boundary conditions imposed d i rect1y on nodes 71 Fig 4.3 Numerical convergence of the FE solution plane strain extrusion R=2, smooth walls 71 Smooth container walls - velocity field, principal Fig b.b stresses, mesh deformation and strain rate isolines for plane strain direct extrusion R=2 75 Rough container walls - velocity field, principal Fig b.5 stresses, mesh deformation and strain rate isolines for plane strain direct extrusion R=2 73 Smooth container walls - velocity field, principal Fig 4.6 stresses, flow lines and strain rate isolines for plane strain direct extrusion R=10 81 Rough container walls - velocity field, principal Fig 4.7 stresses, flow lines and strain rate isolines for plane strain direct extrusion R=10 82 Fig 4.8 Indirect extrusion - boundary conditions imposed d i rect1y on nodes 84 Fig b.3 Indirect plane strain extrusion R 2 effect of bou boundary conditions at ram for Fig b.8(a) 86 Fig 4.10 Plane strain indirect extrusion R=2. Flow patterns for the two ways of operation 86 Fig 4.11 Smooth container walls - velocity field, principal stresses, mesh deformation and strain rate isolines for plane strain-Jndirect extrusion R=2 89 Fig 4.12 Rough container walls - velocity field, principal stresses, mesh deformation and strain rate isolines for plane strain indirect extrusion R=2 90 vi i Fig 4.13 Rough container walls ^velocity field, principal stresses, flow lines and strain rate isolines for for plane strain indirect extrusion R=10 91 Fig 4.14 Dependence of strain rate, strain and hydrostatic pressure with mode of deformation and frictional conditions. Plane strain extrusion. R =20. Perfectly plastic material 97 Fig 4.15 Detail of principal stresses at the exit. Plane strain indirect extrusion R =20 98 Fig 4.16 Dependence of pressures with ram displacement. Axisymmetric direct and indirect extrusion 104 Fig 4.17 Comparison of FE and Upper Bound results. Axisymmetric direct rod extrusion 105 Fig 4.18 Comparison of FE and Upper Bound results. Axisymmetric indirect rod extrusion 106 Fig 4.19 Strain rate dependence with extrusion ratio. Axisymmetric direct and indirect rod extrusion 108 Fig 4.20 Effect of frictional conditions on flow pattern. Axisymmetric extrusion R=20 107 Fig **.21 Dependence of strain rate with mode of deformation and frictional condition. Axisymmetric extrusion R=20 110 Fig 4.22 Dependence of strain with mode of deformation and frictional condition. Axisymmetric extrusion R=20 111 Fig 4.23 Dependence of hydrostatic pressure with deformation and frictional condition. Axsymmetric extrusion R=20 112 Chapter Five Fig 5-1 Hot working behaviour of two Al alloys 119 Fig 5.2 Solution for the one-dimensional thermal model 131 Fig 5-3 Mesh geometry for FE analysis of direct rod extrusion R=12.4 137 Fig 5.4 Thermal boundary conditions 137 Fig 5-5 Direct rod extrusion R=12.4. Effect of heat transfer to container and friction on load-displacement curve 141 Fig 5-6 Effect of both initial billet and container tempe- rature on pressures. R=12.4 142 Fig 5.7 Flow lines for axisymmetric direct rod extrusion R=12.4 144 Fig 5.8 Hot deformation history along flow lines. Direct rod extrusion R=12.4 144 Fig 5.9 Dependence of the temperature rise on initial billet and container temperatures 145 Fig 5.10 Velocity field and principal stress. Direct rod extrusion R=12.4 1i,9 vi i i Fig 5. 11 Strain dependence with ram speed.
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