ISBN: 3540678425
TITLE: Superplastic Flow: Phenomenology and Mechanics
AUTHOR: Padmanabhan/Vasin/Enikeev
TOC:

Introduction 1
1 Phenomenology of Superplastic Flow 5
1.1 Historical 5
1.2 Mechanical Behaviour of Superplastics 6
1.2.1 Mechanical Tests 6
1.2.2 Typical Experimental Results 7
1.2.3 Conditions for Superplastic Flow 8
1.3 Strain Rate Sensitivity of Superplastic Flow 10
1.3.1 Strain Rate Sensitivity Index, m 10
1.3.2 'Universal' Superplastic Curve 12
1.3.3 Stability of Uniaxial Superplastic Flow 14
1.4 Superplasticity from the Point of View of Mechanics 15
1.4.1 On the Definition of Superplasticity 15
1.4.2 On Experimental Studies Concerning Superplasticity 17
1.4.3 On the Presentation of Results Obtained 18
1.4.4 On Some Parameters of Superplastic Flow 20
1.4.4.1 Range of Optimal Flow 20
1.4.4.2 Mechanical Threshold 20
1.4.4.3 Activation Energies 22
1.4.4.4 Structure and Mechanical Response 25
1.4.5 On Stability of Superplastic Flow 26
2 Mechanics of Solids 29
2.1 The Subject 30
2.2 Basic Concepts 33
2.2.1 Concept of a Continuum 33
2.2.2 Stress, Strain and Strain Rate States 35
2.3 General Laws and Boundary Value Problems 38
2.4 Mathematical Models of Materials 40
2.4.1 Typical Models for Describing Mechanical Behaviour 40
2.4.2 Mechanical Models/Analogues 42
2.4.3 Theories of Plasticity 49
2.4.4 Theories of Creep 57

2.4.4.1 Phenomenology of Creep 57
2.4.4.2 Internal Variable Approach 63
2.5 Experiments in Mechanics 65
2.5.1 Mechanical Tests on Materials 65
2.5.2 Influence of Testing Machine 66
3 Constitutive Equations for Superplastics 69
3.1 Basic Requirements of Constitutive Equations 69
3.2 Phenomenological Constitutive Equations 70
3.2.1 Standard Power Law 71
3.2.2 Polynomial Models 74
3.2.3 Mechanical Modelling 76
3.2.3.1 Generalised Maxwell Body 76
3.2.3.2 Generalised Bingham Body 82
3.2.3.3 Mechanical Threshold: Analyses of Karim and Murty 85
3.2.3.4 Smirnov's Mechanical Analogue 90
3.2.3.5 Models of Murty-Banerjee and Zehr-Backofen 91
3.2.3.6 Combinations of Non-Linear Viscous Elements 91
3.2.4 Smirnov's Model 99
3.2.5 Anelasticity 101
3.2.6 Kinks on the Load Relaxation Curves 103
3.2.7 Mechanistic Model 105
3.2.8 Activation Energies 105
3.3 Physical Constitutive Equations 111
3.3.1 Classical Models 112
3.3.2 Modem Theories 114
3.3.2.1 Model of Ghosh 114
3.3.2.2 Model of Hamilton 115
3.3.2.3 The Model of Pschenichniuk-Astanin-Kaibyshev 116
3.3.2.4 The Model of Perevezentsev et al 118
3.4 Construction of Constitutive Equations 119
3.4.1 Common Scheme 119
3.4.2 Model of Padmanabhan and Schlipf 120
3.5 Constitutive Equations in Tensor Form 133
3.5.1 Non-Uniaxial Stress-Strain States 133
3.5.2 Some Tensor Constitutive Equations 137
3.6 Material Constants from Technological Tests 138
3.6.1 Inverse Problems 139
3.6.2 Constant Pressure Forming of a Rectangular Membrane 141
3.6.3 Constant Pressure Forming of a Circular Membrane 146
3.6.4 Model of Padmanabhan and Schlipf 146
4 Boundary Value Problems in Theory of Superplastic Metalworking 149
4.1 General Formulation of the Boundary Value Problem for Metalworking Processes 149
4.1.1 Basic Concepts and Principal Equations 149
4.1.2 Initial and Boundary Conditions 151
4.1.3 Damage Accumulation 157
4.2 Model Boundary Value Problems in Mechanics of Superplasticity 162
4.2.1 Couette Flow of Superplastics 162
4.2.1.1 Newtonian Viscous Liquid 165
4.2.1.2 Shvedov-Bingham Plastic 166
4.2.1.3 Non-Linear Viscous Material 166
4.2.2 Combined Loading of a Cylindrical Rod by Axial Force and Torque 167
4.2.3 Free Bulging of Spherical and Cylindrical Shells 174
4.2.3.1 Free Forming of a Sphere 174
4.2.3.2 Free Forming of an Infinite Cylindrical Shell 176
4.3 Numerical Solving of Boundary Value Problems in Superplasticity 178
4.3.1 Features of Boundary Value Problems in Mechanics of Superplasticity 178
4.3.2 Finite Element Modelling of Superplastic Metalworking Processes 179
4.3.3 Numerical Models of Superplastic Sheet Forming Processes 185
4.3.3.1 Principal Equations of Membrane Theory 186
4.3.3.2 Numerical Solutions of the Principal Equations of Membrane Theory 188
5 Mathematical Modelling of Superplastic Metalworking Processes 195
5.1 Modelling of Superplastic Bulk Forming Processes 195
5.1.1 General Comments 195
5.1.2 Compression of a Disc using Platens 197
5.1.3 Forging of a Disc by Rotating Dies 199
5.1.3.1 Formulation of the Simplified Boundary Value Problem 199
5.1.3.2 Solving the Simplified Boundary Value Problem 201
5.1.3.3 Analysis of the Solution Obtained 204
5.1.4 Extrusion 205
5.1.5 Die-less Drawing 206
5.1.6 Roll Forming Processes 208
5.1.7 Clutching 213
5.2 Modeling of Sheet Metal Processes 213
5.2.1 Simplifications in Modelling SPF and SPF/DB Processes 215
5.2.2 Main Challenges in Modelling SPF and SPF/DB Processes 216
5.2.3 SPF of Hemispherical Domes 217
5.2.3.1 Finite Strain Behaviour 218
5.2.3.2 Jovane's Model 219
5.2.3.3 Geometric/Kinematic Models 221
5.2.3.4 Model of Cornfield-Johnson and its Modifications 225
5.2.3.5 Holt's Model and its Modifications 226
5.2.4 Free Forming of Spherical Vessels 228
5.2.4.1 Description of the Process 228
5.2.4.2 Mathematical Model 228
5.2.4.3 Wrinkling in Superplastic Forming 230
5.2.5 SPF of a Long Rectangular Membrane 232
5.2.5.1 Thickness Distribution 232
5.2.5.2 Pressure -Time Cycle 234
5.2.5.3 Comparison with Experimental Results 236
5.2.6 Estimating Strain in SPF and SPF/DB Processes 241
5.3 Deformation Processing of Materials 243
5.3.1 General Notes 243
5.3.2 Torsion under Pressure and ECA Extrusion 244
5.3.3 Thermomechanical Conditions for Grain Refinement 246
5.3.4 On Some Principles of Structure Refinement 247
6 Problems and Perspectives 251
6.1 Influence of Strain History on Evolution of Structure 253
6.2 Constitutive Equations Including Structural Parameters 258
6.3 The Concept of Database 'TMT-Structure-Properties' 262
6.4 Challenges in Mechanics of Superplasticity 265
6.4.1 Experimental Superplasticity 265
6.4.2 Constitutive Equations 267
Appendix A: Finite Strain Kinematics of Solids 269
A.1 Basic Concepts 269
A.2 Theory of Deformations 272
A.2.1 Strain Tensors 272
A.2.2 Geometrical Sense of Strain Tensor Components 273
A.2.3 Method of Determining the Principal Components of a Srain Tensor 274
A.2.4 Volumetric and Deviatoric Parts of Strain Tensors 276
A.3 Strain Rate Tensor 277
A.3.1 Covariant Components of Strain Tensor 277
A.3.2 Distortion and Spin Tensors 278
A.3.3 Strain Rate Tensor Invariants 279
A.3.4 Volumetric and Deviatoric Parts of the Strain Rate Tensor 280
A.3.5 On Some Scalar Characteristics of a Deformed State 281
Appendix B: Kinematics of Some Simple Deformation Modes 283

B.1 Tension/Compression of a Cylindrical Rod 283
B.2 Simple Shear 291
B.3 Pure Shear 295
B.4 Bulging of a Sphere 300
B.5 Finite Strain Kinematics under Combined Loading of a Cylindrical Rod by Axial Force and Torque 302
Appendix C: On Dimensional Analysis 311
C.1 Basic Concepts 311
C.2 Viscous Flow 313
C.3 Non-Newtonian Flow 315
C.4 Superplastic Flow 316
C.5 Dimensionless Parameters for the Boundary Value Problem of Superplasticity 316
C.6 Physical Modelling of Superplastics 323
Appendix D: Group Properties of Thermoviscoplasticity 325
D.1 About Single-Parameter Groups of Transforms 325
D.2 Applications of Group Methods in Superplasticity 328
References 331
Index 359
END
