Plasticity Fundamentals and Applications

Plasticity Fundamentals and Applications
اسم المؤلف
P.M. Dixit, U.S. Dixit
التاريخ
8 سبتمبر 2021
المشاهدات
65
التقييم
(لا توجد تقييمات)
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Plasticity Fundamentals and Applications
P.M. Dixit, U.S. Dixit
Contents
Preface .xv
Authors xvii

  1. Solid Mechanics and Its Applications 1
    1.1 Introduction .1
    1.2 Continuum Hypothesis 2
    1.3 Elasto-Plastic Solids 4
    1.4 Applications of Solid Mechanics 5
    1.5 Scope of This Textbook .8
    Exercises 8
  2. Review of Algebra and Calculus of Vectors and Tensors 9
    2.1 Introduction .9
    2.2 Index Notations . 10
    2.3 Kronecker Delta and Levy-Civita Symbols . 16
    2.4 Vectors 20
    2.4.1 Norm of a Vector 21
    2.4.2 Addition of Vectors 24
    2.4.3 Dot Product .25
    2.4.4 Cross Product .25
    2.4.5 Derivative of a Vector Function .27
    2.4.6 Gradient of a Scalar Field 27
    2.4.7 Divergence and Curl of a Vector Field 30
    2.4.8 Green’s Theorem in a Plane 33
    2.4.9 Divergence Theorem of Gauss .35
    2.4.10 Integral Theorem of Stokes . 37
    2.5 Transformation Rules for Vector Components under the
    Rotation of Cartesian Coordinate System . 41
    2.6 Tensors 44
    2.6.1 Transformation Rules for Tensor Components under
    the Rotation of Cartesian Coordinate System 44
    2.6.2 Contraction and Quotient Laws 47
    2.6.3 Some Important Definitions and Properties of Tensor 48
    2.6.4 Eigenvalues of a Tensor . 51
    2.6.5 Polar Decomposition of Tensors 55
    2.6.6 Tensor Calculus 60
    2.6.7 Divergence Theorem . 62
    2.6.8 Stokes’ Theorem . 62
    2.6.9 Norm of a Tensor . 62viii Contents
    2.7 Tensors and Vectors in Curvilinear Coordinates .69
    2.7.1 Scale Factors for Cylindrical and Spherical
    Coordinates . 70
    2.7.2 Gradient of a Vector .72
    2.7.3 Divergence of a Vector .73
    2.7.4 Laplacian of a Scalar 75
    2.7.5 Curl of a Vector . 76
    2.7.6 Volume of an Infinitesimal Element .78
    Exercises 79
  3. Stress .85
    3.1 Introduction .85
    3.2 Stress at a Point 87
    3.3 Surface Forces and Body Forces 90
    3.4 Momentum Balance Laws .92
    3.5 Theorem of Virtual Work .94
    3.6 Cauchy’s Theorem .95
    3.7 Transformation of Stress Components . 103
    3.8 Stresses on an Oblique Plane . 105
    3.9 Principal Stresses 107
    3.10 Maximum Shear Stress 111
    3.11 Octahedral Stresses 113
    3.12 Hydrostatic and Deviatoric Stresses 114
    3.13 Mohr’s Circle 116
    3.13.1 Two-Dimensional Case . 116
    3.13.2 Three-Dimensional Case 120
    Exercises 120
  4. Measures of Deformation and Rate of Deformation 127
    4.1 Introduction . 127
    4.2 Deformation . 127
    4.2.1 Linear Strain Tensor 130
    4.2.2 Infinitesimal Rotation Tensor . 136
    4.3 Deformation Gradient 139
    4.4 Green Strain Tensor 144
    4.5 Almansi Strain Tensor 148
    4.6 Logarithmic Strain Tensor . 152
    4.7 Strain–Displacement Relation in Curvilinear Coordinate 153
    4.8 Transformation of Strain Components 156
    4.9 Principal Strains 158
    4.10 Maximum Shear Strain 158
    4.11 Octahedral Strain 159
    4.12 Volumetric Strain 162
    4.13 Mean and Deviatoric Strain . 162
    4.14 Mohr’s Circle for Strain 163Contents ix
    4.15 Incremental Strain Tensor 165
    4.15.1 Introduction 165
    4.15.2 Incremental Linear Strain Tensor 166
    4.15.3 Incremental Infinitesimal Rotation Tensor 168
    4.16 Material and Local Time Derivative . 169
    4.17 Rate of Deformation Tensor . 172
    4.18 Spin Tensor . 176
    4.19 On Relation between Incremental Strain and Strain
    Rate Tensors . 177
    4.20 Compatibility Conditions 178
    Exercises 180
  5. Incremental and Rate Type of Elastic–Plastic Constitutive
    Relations for Isotropic Materials, Objective Incremental Stress
    and Stress Rate Measures . 187
    5.1 Introduction . 187
    5.2 Elastic Stress–Strain Relations for Small Deformation . 188
    5.2.1 One-Dimensional Experimental Observations . 188
    5.2.2 Generalized (i.e. Three-Dimensional) Stress–Strain
    Relations 190
    5.2.3 Stress–Strain Relations for Isotropic Materials . 191
    5.3 Experimental Observations on Elastic–Plastic Behavior 194
    5.3.1 1-D Experimental Observations on Plasticity 195
    5.3.1.1 Elastic Region 197
    5.3.1.2 Yield Stress 197
    5.3.1.3 Plastic Region 197
    5.3.1.4 Strain Hardening 198
    5.3.1.5 Temperature Softening 200
    5.3.1.6 Viscoplasticity .200
    5.3.1.7 Isochoric Deformation .200
    5.3.1.8 Large Deformation .200
    5.3.1.9 Hysteresis 202
    5.3.1.10 Bauschinger Effect . 202
    5.3.1.11 Effect of Hydrostatic Stress on Yielding .203
    5.3.1.12 Anisotropy 203
    5.4 Criteria for Initial Yielding of Isotropic Materials .204
    5.4.1 von Mises Yield Criterion .204
    5.4.2 Tresca Yield Criterion 209
    5.4.3 Geometric Representation of Yield Criteria . 211
    5.4.4 Convexity of Yield Surfaces 214
    5.4.5 Experimental Validation . 214
    5.5 Modeling of Isotropic Hardening or Criterion for
    Subsequent Isotropic Yielding 215
    5.5.1 Strain-Hardening Hypothesis for Mises Material 217
    5.5.2 Work-Hardening Hypothesis for Mises Material . 219x Contents
    5.5.3 Criterion for Subsequent Yielding for Mises Material
    Based on Strain-Hardening Hypothesis 219
    5.5.4 Experimental Validation of Isotropic Hardening .223
    5.6 Elastic–Plastic Stress–Strain and Stress–Strain Rate
    Relations for Isotropic Materials .223
    5.6.1 Drucker’s Postulate for Stable Plastic Material 224
    5.6.2 Associated Flow Rule 228
    5.6.3 Elastic–Plastic Incremental Stress–Strain Relation for
    the Mises Material .233
    5.6.4 Elastic–Plastic Stress–Strain Rate Relation for the
    Mises Material 234
    5.6.5 Viscoplasticity and Temperature Softening . 237
    5.7 Objective Incremental Stress and Objective Stress
    Rate Tensors .238
    5.7.1 Relation between Cauchy Stress Tensors When the
    Increment Is Pure Rotation . 240
    5.7.2 Piola–Kirchoff Stress Tensors . 242
    5.7.3 Increment of Second Piola–Kirchoff Stress Tensor
    (Objective Incremental Stress Tensor) .244
    5.7.4 Relation between Finite and Infinitesimal
    Incremental Rotation Tensors for Small Increment 246
    5.7.5 Jaumann Stress Tensor (Objective Stress Rate Tensor) . 247
    5.8 Unloading Criterion 248
    Exercises 250
  6. Eulerian and Updated Lagrangian Formulations 253
    6.1 Introduction .253
    6.2 Equation of Motion in Terms of Velocity Derivatives 254
    6.3 Incremental Equation of Motion .255
    6.4 Eulerian Formulation .256
    6.4.1 Governing Equations (Elasto-Plastic Material) . 257
    6.4.2 Governing Equations (Rigid-Plastic Material) 258
    6.4.3 Boundary Conditions 260
    6.4.4 Initial Conditions . 261
    6.5 Example of Eulerian Formulation: A Wire Drawing Problem . 261
    6.5.1 Inlet and Exit Boundaries AB and EF . 262
    6.5.2 Stress-Free Boundaries BC and DE .263
    6.5.3 Plane of Symmetry AF 263
    6.5.4 Die Interface CD .263
    6.5.5 Location of Plastic Boundaries .265
    6.6 Updated Lagrangian Formulation 266
    6.6.1 Governing Equations 266
    6.6.2 Boundary Conditions 268
    6.6.2.1 Initial Conditions . 269
    6.6.2.2 Updating Scheme . 269Contents xi
    6.7 Example on Updated Lagrangian Formulation: Forging of a
    Cylindrical Block . 269
    6.7.1 Stress-Free Boundary BC 270
    6.7.2 Plane of Symmetry DC . 271
    6.7.3 Plane of Symmetry AD . 271
    6.7.4 Platen Interface AB 271
    Exercises 272
  7. Calculus of Variations and Extremum Principles 275
    7.1 Introduction . 275
    7.2 Functional 278
    7.3 Extremization of a Functional .283
    7.3.1 Functional Containing the Form F(x,y,y′) .283
    7.3.2 Alternate Form of Euler–Lagrange Equation 289
    7.3.3 Functional Containing the Form
    F x = ( , , y y y 1 1, , , 2 2 y y ., n n , ) y . 291
    7.3.4 Functional Containing the Function of n
    Independent Variables 293
    7.3.5 Functional Dependent on the Functions and Its
    Derivatives up to Order n . 296
    7.4 Solution of Extremization Problems Using δ Operator .299
    7.4.1 Variational Operator 299
    7.4.2 Properties of Variational Operator 300
    7.4.3 Converting Variational Form to Differential Equation 302
    7.5 Obtaining Variational Form from a Differential Equation .305
    7.6 Principle of Virtual Work . 312
    7.7 Principle of Minimum Potential Energy . 314
    7.8 Solution of Variational Problems by Ritz Method 315
    Exercises 317
  8. Two-Dimensional and Axisymmetric Elasto-Plastic Problems 323
    8.1 Introduction . 323
    8.2 Symmetric Beam Bending of a Perfectly Plastic Material
    (1-D Problem) . 323
    8.2.1 Pure Bending 324
    8.2.1.1 Elastic Analysis . 324
    8.2.1.2 Plastic Analysis . 327
    8.2.2 Bending in the Presence of Shear Force 331
    8.3 Hole Expansion in an Infinite Plate (Plane Stress and
    Axisymmetric Problem) .336
    8.3.1 Initial Yielding 337
    8.3.2 Elasto-Plastic Analysis for a Perfectly Plastic Material .339
    8.3.2.1 Stresses in the Elastic Region .339
    8.3.2.2 Stresses in the Plastic Region .339
    8.3.3 Elasto-Plastic Analysis for a Hardening Material .343
    8.4 Analysis of Plastic Deformation in the Flange of Circular
    Cup during Deep Drawing Process (Plane Stress and
    Axisymmetric Problem) . 351
    8.4.1 Determination of Stresses .353
    8.4.2 Determination of Strains 355
    8.4.2.1 Determination of Logarithmic Hoop Strain .356
    8.4.2.2 Determination of Logarithmic Thickness
    Strain 359
    8.5 Necking of a Cylindrical Rod 361
    8.5.1 Analysis in the Plane of Symmetry (z = 0) .363
    8.5.1.1 Simplification of Differential Equation .365
    8.5.1.2 Solution of the Modified Differential Equation . 369
    Exercises 371
    Appendix A 380
    Appendix B . 382
  9. Contact Mechanics .385
    9.1 Introduction .385
    9.2 Hertz Theory 386
    9.2.1 Geometry of Unstressed Surface in the
    Region of Contact . 387
    9.2.2 Boussinesq Solution . 392
    9.2.3 Pressure and Deflections in the Contact Region . 395
    9.2.4 Two Spheres in Contact . 397
    9.2.5 Two Cylinders in Contact along a Line Parallel to
    Their Axes .400
    9.2.6 Alternate Derivation for the Contact between
    Two Cylinders .402
    9.2.7 Stresses in Contact Problem .405
    9.3 Elastic–Plastic Indentation .407
    9.3.1 Solution of Flat Plate Indentation Problem by Upper
    Bound Method .409
    9.3.1.1 Power Dissipation along AB . 411
    9.3.1.2 Power Dissipation along BC . 411
    9.3.1.3 Power Dissipation along BD . 412
    9.3.1.4 Power Dissipation along CD . 412
    9.3.1.5 Power Dissipation along DE . 412
    9.3.2 Solution of Flat Plate Indentation by Slip Line
    Field Method . 413
    9.3.3 Solution of Flat Plate Indentation by
    Numerical Methods . 416
    9.4 Cavity Model . 418
    9.4.1 Determination of Elastic–Plastic Boundary Radius .422
    9.4.2 Determination of Plastic Strain 424
    9.4.3 Typical Results 425
    9.5 Sliding of Elastic–Plastic Solids 427
    9.6 Rolling Contact 428
    9.7 Principle of Virtual Work and Discretization of
    Contact Problems 431
    Exercises 433
  10. Dynamic Elasto-Plastic Problems . 437
    10.1 Introduction . 437
    10.2 Longitudinal Stress Wave Propagation in a Rod (1-D Problem) 437
    10.2.1 Method of Characteristics 439
    10.2.2 Conditions at the Surfaces of Discontinuity in
    Wave Propagation 441
    10.2.3 Elastic Solution of 1-D Wave Equation 442
    10.2.4 1-D Wave Equation for Unloading 444
    10.2.5 Plastic Solution of 1-D Wave Equation in Rod
    Impacted against Rigid Support 445
    10.3 Taylor Rod Problem (Impact of Cylindrical Rod against
    Flat Rigid Surface, 1-D Problem) .450
    10.3.1 Governing Equations 452
    10.3.1.1 Kinematic Relations . 452
    10.3.1.2 Equation of Motion 452
    10.3.1.3 Volume Constancy Condition 453
    10.3.2 Determination of x as a Function of e .453
    10.3.3 Determination of h as a Function of e .456
    10.3.4 Determination of t as a Function of e 457
    10.3.5 Energy Method .459
    Exercises 459
  11. Continuum Damage Mechanics and Ductile Fracture . 461
    11.1 Introduction . 461
    11.2 Motivation 462
    11.2.1 Failure of the Titanic 462
    11.2.2 Failure of Liberty Ships . 462
    11.2.3 Failure of Comet Passenger Aircraft . 462
    11.2.4 Failure of the Space Shuttle Challenger 463
    11.3 Objective and Plan of the Chapter 463
    11.4 Classification of Fracture .464
    11.5 Global and Local Approaches to Fracture .465
    11.5.1 Limitations of Global and Local Approaches
    to Fracture .466
    11.6 Ductile Fracture . 467
    11.6.1 Void Nucleation or Initiation 468
    11.6.2 Void Growth . 470
    11.6.2.1 Analytical Models for Void Growth 470
    11.6.3 Void Coalescence 472
    11.7 Models of Fracture Initiation . 474
    11.7.1 Porous Plasticity Model (Gurson and GTN Model) 475
    11.7.2 CDM-Based Model: Review of Literature 477
    11.7.3 Other Models of Fracture Initiation 478
    11.8 Thermodynamics of Continuum 481
    11.8.1 Thermodynamic Process with Internal Variables .482
    11.8.2 Thermo-Elastic–Plastic Process .483
    11.9 Continuum Damage Mechanics .485
    11.9.1 Length Scales of Damage 485
    11.9.2 Representative Volume Element 487
    11.9.3 Requirements of Damage Modeling .488
    11.9.4 Definition of a Scalar Damage Variable 488
    11.9.5 Effective Stress Concept 490
    11.9.6 Crack Initiation Criterion 491
    11.9.7 Strain Equivalence Principle 491
    11.9.8 Elastic Strain Energy Equivalence Principle 492
    11.9.9 Thermodynamic Force Corresponding to Damage 493
    11.9.10 Constitutive Equations for Thermo-Elasto–Plastic
    Process in a Damaged Material . 497
    11.9.11 Damage Growth Laws 499
    11.9.12 Microcrack Closure Effect 502
    11.10 Techniques for Damage Measurement 504
    11.11 Application of a CDM Model 506
    11.11.1 Procedure for Determining Damage Law
    Coefficients in Equation 11.122 .507
    11.11.2 Tensile Testing and Ductile Fracture of
    Cylindrical Specimen 508
    11.12 Closure and Further Reading 513
    Exercises 513
  12. Plastic Anisotropy 515
    12.1 Introduction . 515
    12.2 Normal and Planar Anisotropy 515
    12.3 Hill’s Anisotropic Yield Criteria . 519
    12.4 Plane Stress Anisotropic Yield Criterion of Barlat and Lian 529
    12.5 Three-Dimensional Anisotropic Yield Criteria of Barlat
    and Coworkers .533
    12.6 Plane Strain Anisotropic Yield Criterion . 537
    12.7 Constitutive Relations for Anisotropic Materials 539
    12.8 Kinematic Hardening .542
    Exercises 545
    References

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