Modeling of metal forming and machining processes : by finite element and soft computing methods

The use of computational techniques is increasing day by day in the manufacturing sector. Process modeling and optimization with the help of computers can reduce expensive and time consuming experiments for manufacturing good quality products. Metal forming and machining are two prominent manufacturing processes. Both of these processes involve large deformation of elasto-plastic materials due to applied loads. In metal forming, the material is plastically deformed without causing fracture. On the other hand, in machining, the material is deformed till fracture, in order to remove material in the form of chips. To understand the physics of metal forming and machining processes, one needs to understand the kinematics of large deformation (dependence of deformation and its rate on displacement) as well as the constitutive behavior of elasto-plastic materials (dependence of internal forces on deformation and its rate). Once the physics is understood, these phenomena have to be converted to mathematical relations in the form of differential equations. The interaction of the work-piece with the tools/dies and other surroundings also needs to be expressed in a mathematical form (known as the boundary and initial conditions). In this book, the first four chapters essentially discuss the physics of metal forming and machining processes. The physical behavior of the work-piece during the processes is modeled in the form of differential equations and boundary and initial conditions.

Author Notes

Prof. P. M. Dixit has been actively working in the area of metal forming, machining and non-traditional machining for the past 20 years. He has published extensively in leading international journals and carried out projects in the area of metal forming and large deformation. He also teaches Metal Forming, Plasticity and FEM to postgraduate and senior undergraduate students. Prof. P.M. Dixit obtained his bachelors degree in Aeronautical Engineering from the Indian Institute of Technology, Kharagpur, in 1974. He was awarded a silver medal for securing the first rank in the Department. Subsequently, he obtained his doctoral degree in Mechanics from the University of Minnesota, Minneapolis, U.S.A, in 1979. After receiving his Ph.D. degree, Prof. Dixit taught at the Aeronautical Engineering Department of the Indian Institute of Technology, Kharagpur for 4 years (1980-1984). Since 1984, Prof. Dixit has been teaching at the Mechanical Engineering Department of the Indian Institute of Technology, Kanpur.

Prof. U. S. Dixit has more than a decade's experience in carrying out research in the area of metal forming and machining. Apart from FEM, he uses fuzzy set theory and neural networks in his research. Before taking up a research career, he worked for four years as a machine tool designer in HMT Ltd. Pinjore, India. He has a number of publications, some of them jointly with Prof. P.M. Dixit. Prof. U. S. Dixit is currently a Professor of Mechanical Engineering at the Indian Institute of Technology in Guwahati. Prof. U. S. Dixit obtained his bachelors degree in Mechanical Engineering from the Indian Institute of Technology, Roorkee in 1987. He gained his M. Tech in Mechanical Engineering and his Ph.D. in Mechanical Engineering from the Indian Institute of Technology, Kanpur, in 1993 and 1998 respectively.

1 Metal Forming and Machining Processes	p. 1
1.1 Introduction	p. 1
1.2 Metal Forming	p. 2
1.2.1 Bulk Metal Forming	p. 2
1.2.2 Sheet Metal Forming Processes	p. 17
1.3 Machining	p. 23
1.3.1 Turning	p. 24
1.3.2 Milling	p. 28
1.3.3 Some Other Machining Processes	p. 30
1.4 Summary	p. 31
1.5 References	p. 31
2 Review of Stress, Linear Strain and Elastic Stress-Strain Relations	p. 33
2.1 Introduction	p. 33
2.2 Index Notation and Summation Convention	p. 35
2.3 Stress	p. 41
2.3.1 Stress at a Point	p. 41
2.3.2 Analysis of Stress at a Point	p. 52
2.3.3 Equation of Motion	p. 61
2.4 Deformation	p. 64
2.4.1 Linear Strain Tensor	p. 65
2.4.2 Analysis of Strain at a Point	p. 75
2.4.3 Compatibility Conditions	p. 82
2.5 Material Behavior	p. 84
2.5.1 Elastic Stress-Strain Relations for Small Deformations	p. 85
2.6 Summary	p. 93
2.7 References	p. 94
3 Classical Theory of Plasticity	p. 95
3.1 Introduction	p. 95
3.2 One-Dimensional Experimental Observations on Plasticity	p. 97
3.3 Criteria for Initial Yielding of Isotropic Materials	p. 107
3.3.1 von Mises Yield Criterion	p. 108
3.3.2 Tresca Yield Criterion	p. 110
3.3.3 Geometric Representation of Yield Criteria	p. 111
3.3.4 Convexity of Yield Surfaces	p. 114
3.3.5 Experimental Validation	p. 115
3.4 Incremental Strain and Strain Rate Measures	p. 121
3.4.1 Incremental Linear Strain Tensor	p. 121
3.4.2 Strain Rate Tensor	p. 125
3.4.3 Relation Between Incremental Linear Strain Tensor and Strain Rate Tensor	p. 130
3.5 Modeling of Isotropic Hardening or Criterion for Subsequent Isotropic Yielding	p. 134
3.5.1 Strain Hardening Hypothesis	p. 136
3.5.2 Work Hardening Hypothesis	p. 138
3.5.3 Experimental Validations	p. 138
3.6 Plastic Stress-Strain and Stress-Strain Relations for Isotropic Materials	p. 141
3.6.1 Associated Flow Rule	p. 143
3.6.2 Elastic-Plastic Incremental Stress-Strain Relations for Mises Material	p. 151
3.6.3 Elastic-Plastic Incremental Stress-Strain Rate Relation for Mises Material	p. 153
3.6.4 Viscoplasticity and Temperature Softening	p. 157
3.7 Objective Stress Rate and Objective Incremental Stress Tensors	p. 161
3.7.1 Jaumann Stress Rate and Associated Objective Incremental Stress Tensor	p. 163
3.8 Unloading Criterion	p. 168
3.9 Eulerian and Updated Lagrangian Formulations for Metal Forming Processes	p. 170
3.9.1 Equation of Motion in Terms of Velocity Derivatives	p. 170
3.9.2 Incremental Equation of Motion	p. 172
3.9.3 Eulerian Formulation for Metal Forming Problems	p. 173
3.9.4 Updated Lagrangian Formulation for Metal Forming Problems	p. 182
3.10 Eulerian Formulation for Machining Processes	p. 188
3.11 Summary	p. 192
3.12 References	p. 193
4 Plasticity of Finite Deformation and Anisotropic Materials and Modeling of Fracture and Friction	p. 195
4.1 Introduction	p. 195
4.2 Kinematics of Finite Deformation and Rotation	p. 197
4.3 Constitutive Equation for Eulerian Formulation When the Rotation Is Not Small	p. 207
4.3.1 Solution Procedure	p. 210
4.4 Kinematics of Finite Incremental Deformation and Rotation	p. 212
4.5 Constitutive Equation for Updated Lagrangian Formulation for Finite Incremental Deformation and Rotation	p. 219
4.6 Anisotropic Initial Yield Criteria	p. 223
4.6.1 Hill's Anisotropic Yield Criteria	p. 226
4.6.2 Plane Stress Anisotropic Yield Criterion of Barlat and Lian	p. 227
4.6.3 A Three-Dimensional Anisotropic Yield Criterion of Barlat and Co-workers	p. 229
4.6.4 A Plane Strain Anisotropic Yield Criterion	p. 236
4.7 Elastic-Plastic Incremental Stress-Strain and Stress-Strain Rate Relations for Anisotropic Materials	p. 239
4.7.1 Elastic-Plastic Incremental Stress-Strain Relation for Anisotropic Materials	p. 239
4.7.2 Elastic-Plastic Incremental Stress-Strain Rate Relation for Anisotropic Materials	p. 243
4.8 Kinematic Hardening	p. 247
4.9 Modeling of Ductile Fracture	p. 252
4.9.1 Porous Plasticity Model of Berg and Gurson	p. 252
4.9.2 Void Nucleation, Growth and Coalescence Model (Goods and Brown, Rice and Trace and Thomason Model)	p. 253
4.9.3 Continuum Damage Mechanics Models	p. 257
4.9.4 Phenomenological Models	p. 262
4.10 Friction Models	p. 265
4.10.1 Wanheim and Bay Friction Model	p. 266
4.11 Summary	p. 268
4.12 References	p. 269
5 Finite Element Modeling of Metal Forming Processes Using Eulerian Formulation	p. 273
5.1 Introduction	p. 273
5.2 Background of Finite Element Method	p. 274
5.2.1 Pre-processing	p. 274
5.2.2 Developing Elemental Equations	p. 285
5.2.3 Assembly Procedure	p. 292
5.2.4 Applying Boundary Conditions	p. 295
5.2.5 Solving the System of Equations	p. 296
5.2.6 Post-processing	p. 296
5.3 Formulation of Plane-Strain Metal Forming Processes	p. 297
5.3.1 Governing Equations and Boundary Conditions	p. 298
5.3.2 Non-Dimensionalization	p. 301
5.3.3 Weak Formulation	p. 302
5.3.4 Finite Element Formulation	p. 304
5.3.5 Application of Boundary Conditions	p. 311
5.3.6 Estimation of Neutral Point	p. 313
5.3.7 Formulation for Strain Hardening	p. 315
5.3.8 Modification of Pressure Field at Each Iteration	p. 316
5.3.9 Calculation of Secondary Variables	p. 318
5.3.10 Some Numerical Aspects	p. 319
5.3.11 Typical Results and Discussion	p. 320
5.4 Formulation of Axisymmetric Metal Forming Processes	p. 322
5.5 Formulation of Three-Dimensional Metal Forming Processes	p. 331
5.6 Incorporation of Anisotropy	p. 331
5.7 Elasto-Plastic Formulation	p. 334
5.8 Summary	p. 341
5.9 References	p. 341
6 Finite Element Modeling of Metal Forming Processes Using Updated Lagrangian Formulation	p. 345
6.1 Introduction	p. 345
6.2 Application of Finite Element Method to Updated Lagrangian Formulation	p. 347
6.2.1 Governing Equations	p. 347
6.2.2 Integral Form of Equilibrium Equation	p. 349
6.2.3 Finite Element Formulation	p. 351
6.2.4 Evaluation of the Derivative	p. 356
6.2.5 Iterative Scheme	p. 365
6.2.6 Determination of Stresses	p. 368
6.2.7 Divergence Handling Techniques	p. 371
6.3 Modeling of Axisymmetric Open Die Forging by Updated Lagrangian Finite Element Method	p. 372
6.3.1 Domain and Boundary Conditions	p. 374
6.3.2 Cylindrical Arc Length Method for Displacement Control Problems	p. 377
6.3.3 Friction Algorithm	p. 380
6.3.4 Convergence Study and Evaluation of Secondary Variables	p. 382
6.3.5 Validation of the Finite Element Formulation	p. 382
6.3.6 Typical Results	p. 384
6.3.7 Residual Stress Distribution	p. 388
6.3.8 Damage Distribution, Hydrostatic Stress Distribution and Fracture	p. 393
6.4 Modeling of Deep Drawing of Cylindrical Cups by Updated Lagrangian Finite Element Method	p. 396
6.4.1 Domain and Boundary Conditions	p. 399
6.4.2 Contact Algorithm	p. 405
6.4.3 Typical Results	p. 406
6.4.4 Anisotropic Analysis, Ear Formation and Parametric Studies	p. 408
6.4.5 Optimum Blank Shape	p. 416
6.5 Summary	p. 419
6.6 References	p. 420
7 Finite Element Modeling of Orthogonal Machining Process	p. 425
7.1 Introduction	p. 425
7.2 Domain, Governing Equations and Boundary Conditions for Eulerian Formulation	p. 426
7.2.1 Domain	p. 426
7.2.2 Governing Equations	p. 428
7.2.3 Boundary Conditions	p. 429
7.3 Finite Element Formulation	p. 431
7.3.1 Integral Form	p. 431
7.3.2 Approximations for Velocity Components and Pressure	p. 433
7.3.3 Finite Element Equations	p. 436
7.3.4 Application of Boundary Conditions, Solution Procedure and Evaluation of Secondary Quantities	p. 440
7.4 Results and Discussion	p. 442
7.4.1 Validation of the Formulation	p. 444
7.4.2 Parametric Studies	p. 444
7.4.3 Primary Shear Deformation Zone, Contours of Equivalent Strain Rate and Contours of Equivalent Stress	p. 445
7.5 Summary	p. 447
7.6 References	p. 448
8 Background on Soft Computing	p. 451
8.1 Introduction	p. 451
8.2 Neural Networks	p. 452
8.2.1 Biological Neural Networks	p. 453
8.2.2 Artificial Neurons	p. 454
8.2.3 Perceptron: The Learning Machine	p. 458
8.2.4 Multi-Layer Perceptron Neural Networks	p. 462
8.2.5 Radial Basis Function Neural Network	p. 469
8.2.6 Unsupervised Learning	p. 471
8.3 Fuzzy Sets	p. 472
8.3.1 Mathematical Definition of Fuzzy Set	p. 473
8.3.2 Some Basic Definitions and Operations	p. 474
8.3.3 Determination of Membership Function	p. 476
8.3.4 Fuzzy Relations	p. 480
8.3.5 Extension Principle	p. 481
8.3.6 Fuzzy Arithmetic	p. 482
8.3.7 Fuzzy Sets vs Probability	p. 483
8.3.8 Fuzzy Logic	p. 484
8.3.9 Linguistic Variables and Hedges	p. 484
8.3.10 Fuzzy Rules	p. 486
8.3.11 Fuzzy Inference	p. 486
8.4 Genetic Algorithms	p. 491
8.4.1 Binary Coded Genetic Algorithms	p. 492
8.4.2 Real Coded Genetic Algorithms	p. 497
8.5 Soft Computing vs FEM	p. 498
8.6 Summary	p. 499
8.7 References	p. 500
9 Predictive Modeling of Metal Forming and Machining Processes Using Soft Computing	p. 503
9.1 Introduction	p. 503
9.2 Design of Experiments and Preliminary Study of the Data	p. 504
9.3 Preliminary Statistical Analysis	p. 508
9.3.1 Correlation Analysis	p. 508
9.3.2 Hypothesis Testing	p. 509
9.3.3 Analysis of Variance	p. 515
9.3.4 Multiple Regression	p. 518
9.4 Neural Network Modeling	p. 522
9.4.1 Selection of Training and Testing Data	p. 523
9.4.2 Deciding the Processing Functions	p. 525
9.4.3 Effect of Number of Hidden Layers	p. 525
9.4.4 Effect of Number of Neurons in the Hidden Layers	p. 525
9.4.5 Effect of Spread Parameter in Radial Basis Function Neural Network	p. 526
9.4.6 Data Filtration	p. 528
9.4.7 Lower and Upper Estimates	p. 528
9.5 Prediction of Dependent Variables Using Fuzzy Sets	p. 533
9.6 Prediction Using ANFIS	p. 535
9.7 Computation with Fuzzy Variables	p. 539
9.8 Summary	p. 545
9.9 References	p. 546
10 Optimization of Metal Forming and Machining Processes	p. 549
10.1 Introduction	p. 549
10.2 Optimization Problems in Metal Forming	p. 550
10.2.1 Optimization of Roll Pass Scheduling	p. 551
10.2.2 Optimization of Rolls	p. 554
10.2.3 Optimization of Wire Drawing and Extrusion	p. 554
10.2.4 A Brief Review of Other Optimization Studies in Metal Forming	p. 556
10.3 Optimization Problems in Machining	p. 559
10.3.1 A Brief Review of Optimization of Machining Processes	p. 559
10.3.2 Optimization of Multipass Turning Process	p. 563
10.3.3 Online Determination of Equations for Machining Performance Parameters	p. 569
10.4 Summary	p. 573
10.5 References	p. 573
11 Epilogue	p. 579
11.1 References	p. 583
Index	p. 585

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