Plasticity-Damage Couplings: From Single Crystal to Polycrystalline Materials (eBook)

Preface 6
Contents 9
1 Mathematical Framework 14
1.1 Elements of Vector Algebra 14
1.2 Elements of Tensor Algebra 20
1.2.1 Second-Order Tensors 20
1.2.2 Higher-Order Tensors 34
1.3 Elements of Vector and Tensor Calculus 40
1.4 Elements of the Theory of Tensor Representation 43
1.4.1 Symmetry Transformations and Groups 43
1.4.2 Representation Theorems for Orthotropic Scalar Functions 47
References 48
2 Constitutive Equations for Elastic–Plastic Materials 49
2.1 Stress-Based Formulation of Elastic–Plastic Models 56
2.1.1 Ideal Plasticity 56
2.1.2 Elastic–Plastic Work-Hardening Materials 58
2.1.3 Time Integration Algorithm for Stress-Based Elastic–Plastic Constitutive Models 63
2.2 Strain-Rate-Based Formulation for Elastic–Plastic Models 65
2.2.1 Mathematical Framework 65
2.2.2 Time Integration Algorithm for Strain-Rate-Based Elastic–Plastic Models 67
References 70
3 Plastic Deformation of Single Crystals 73
3.1 Elements of Crystallography 73
3.2 Plastic Deformation Mechanisms in Crystals: Experimental Evidence 81
3.2.1 Crystallographic Slip 81
3.2.2 Deformation Twinning 86
3.3 Yield Criteria for Single Crystals 89
3.3.1 Generalized Schmid Yield Criterion 90
3.3.2 Cazacu et al. [26] Yield Criterion 92
3.3.2.1 Effect of Loading Orientation on Yielding 98
3.3.2.2 Procedure for Identification of the Yield Criterion 100
3.3.3 Application to the Description of Yielding in Cu and Al Single Crystals 102
3.3.3.1 Cu Single Crystal 102
3.3.3.2 Al 5% Cu Single Crystal 106
3.3.4 Application of Cazacu et al. [26] Single Crystal Criterion to Deep Drawing 109
3.4 Modeling of Plastic Anisotropy of Polycrystalline Textured Sheets Based on Cazacu et al. [26] Single Crystal Criterion 117
3.4.1 Analytical Expressions for the Yield Stress and Lankford Coefficients of Ideal Texture Components 120
3.4.1.1 Cube Texture 121
3.4.1.2 Goss Texture /left/{ {{/bf 110}} /right/} /langle {/bf 001} /rangle 124
3.4.1.3 Brass Texture /{ /bar{2}1/bar{1}/} /langle 011 /rangle 126
3.4.1.4 Copper Texture /left/{ {112} /right/} /langle 11/bar{1} /rangle 129
3.4.1.5 Rotated Cube Texture /left/{ {100} /right/} /langle 011 /rangle 130
3.4.2 Prediction of Plastic Anisotropy of Textured Polycrystalline Sheets with Several Texture Components 133
3.4.2.1 Effect of the Spread About Ideal Textures on the Uniaxial Plastic Properties 134
3.4.2.2 Predictions of Anisotropy of Yield Stresses and Lankford Coefficients for Textured Sheets 139
3.4.2.3 Applications to Polycrystalline Al and Steel Sheets 143
References 147
4 Yield Criteria for Isotropic Polycrystals 152
4.1 General Mathematical Requirements 152
4.1.1 General Form of Isotropic Yield Criteria 152
4.1.2 Representation of the Yield Surface of Isotropic Materials in the Octahedral Plane 154
4.2 Yield Criteria for Isotropic Metallic Materials Displaying the Same Response in Tension–Compression 158
4.2.1 Classical Yield Criteria 158
4.2.1.1 von Mises [44] Yield Criterion 158
4.2.1.2 Tresca [42] Yield Criterion 160
4.2.2 Drucker [15] Yield Criterion 164
4.2.3 Hershey–Hosford Yield Criterion 170
4.3 Yield Criteria for Isotropic Metallic Materials Showing Asymmetry Between the Response in Tension–Compression 173
4.3.1 Cazacu and Barlat [8] Yield Criterion 173
4.3.2 Cazacu et al. [9] Isotropic Yield Criterion 180
4.4 Application of the Cazacu et al. [9] Yield Criterion to the Description of Plastic Deformation Under Torsion 188
4.4.1 Monotonic Torsion: Analytical Results 188
4.4.2 F.E. Simulations of Monotonic Free-End Torsion 192
4.4.3 Application to Commercially Pure Al 197
4.5 Cyclic Torsional Loading 199
References 210
5 Yield Criteria for Anisotropic Polycrystals 212
5.1 General Methods for Extending to Anisotropy Yield Criteria for Isotropic Materials 212
5.1.1 Generalized Orthotropic Invariants 213
5.1.2 Generalized Transversely Isotropic Invariants 216
5.2 Orthotropic Generalization of von Mises Isotropic Criterion Due to Hill [22] 217
5.2.1 Yield Stress Anisotropy Predicted by the Hill [22] Criterion 220
5.2.2 Variation of the Lankford Coefficients with the Tensile Loading Direction According to Hill [22] Criterion 228
5.2.3 Comments on the Identification Procedure 228
5.3 Non-quadratic Three-Dimensional Yield Criteria for Materials with the Same Response in Tension–Compression 231
5.3.1 Cazacu and Barlat [11] Orthotropic Criterion 231
5.3.1.1 Predicted Anisotropy in Yield Stresses and Lankford Coefficients 235
5.3.1.2 Extension of Drucker [16] Isotropic Yield Criterion to Transversely Isotropic Materials 238
5.3.2 Cazacu [10] Orthotropic Yield Criterion 239
5.3.2.1 Anisotropy in Lankford Coefficients and Uniaxial Yield Stresses in the Plane (RD, TD) 241
5.3.2.2 Anisotropy in Yield Stresses in the Other Symmetry Planes 244
5.3.3 Explicit Expression of the Barlat et al. [4] Orthotropic Yield Criterion in Terms of Stresses 250
5.3.4 Explicit Expression of the Karafillis and Boyce [28] Orthotropic Yield Criterion in Terms of Stresses 255
5.3.5 Explicit Expression of Yld 2004-18p Orthotropic Yield Criterion in Terms of Stresses 257
5.3.6 Explicit Expression of Yld 2004-13p Orthotropic Yield Criterion in Terms of Stresses 261
5.4 Yield Criteria for Textured Polycrystals with Tension–Compression Asymmetry 262
5.4.1 Orthotropic Yield Criterion of Cazacu and Barlat [13] 263
5.4.2 Orthotropic Yield Criterion of Nixon et al. [36] 268
5.4.2.1 Yielding Formulation 268
5.4.2.2 Applications: Tension, Compression, and Bending of hcp-Ti 272
5.4.3 Orthotropic and Asymmetric Yield Criterion of Cazacu et al. [14] 283
5.4.3.1 Yielding Description 283
5.4.3.2 Applications: Tension, Compression, and Torsion of hcp-Ti and Mg AZ31 288
References 297
6 Strain-Rate-Based Plastic Potentials for Polycrystalline Materials 300
6.1 Isotropic Strain-Rate Plastic Potentials 300
6.1.1 Strain-Rate Potentials for Isotropic Metallic Materials with the Same Response in Tension–Compression 302
6.1.1.1 Exact Duals of the von Mises and Tresca Stress Potentials 302
6.1.1.2 Hershey–Hosford Pseudo-Strain-Rate Potential 305
6.1.1.3 Strain-Rate Potential of Cazacu and Revil-Baudard [7] 308
6.1.2 Strain-Rate Potentials for Isotropic Metallic Materials with Asymmetry Between Tension–Compression 310
6.1.2.1 Exact Dual of the Isotropic Cazacu et al. [5] Stress Potential 310
6.1.2.2 Application to Fixed-End Torsion 318
6.2 Orthotropic Strain-Rate Plastic Potentials 322
6.2.1 Strain-Rate Potentials for Orthotropic Materials with the Same Response in Tension–Compression 322
6.2.1.1 Exact Dual of the Hill [14] Stress Potential 323
6.2.1.2 Orthotropic Strain-Rate Potential of Barlat et al. [2]: SRP93 328
6.2.1.3 Orthotropic Strain-Rate Potential of Barlat and Chung [4]: SRP2004-18p 331
6.2.2 Exact Dual of the Orthotropic Cazacu et al. [5] Stress Potential 336
References 345
7 Plastic Potentials for Isotropic Porous Materials: Influence of the Particularities of Plastic Deformation on Damage Evolution 347
7.1 Kinematic Homogenization Framework for Development of Plastic Potentials for Porous Metallic Materials 349
7.2 Constitutive Models for Porous Isotropic Metallic Materials with Incompressible Matrix Governed by an Even Yield Function 351
7.2.1 General Properties of the Yield Surface of Porous Metallic Materials Containing Spherical Voids in an Incompressible Matrix Governed by an Even Yield Function 352
7.2.2 Velocity Field Compatible with Uniform Strain-Rate Boundary Conditions 354
7.2.2.1 Rice and Tracey [58] Velocity Field 355
7.2.3 Porous Materials with von Mises Matrix 357
7.2.3.1 Gurson [30] Plastic Potentials 357
7.2.3.2 Modified Versions of Gurson [30] Criterion 361
7.2.3.3 Combined Effects of Mean Stress and Third-Invariant on the Mechanical Response According to Cazacu et al. [19] Plastic Potential 368
7.2.3.4 Void Growth and Collapse According to Cazacu et al. [19] Model and F.E. Unit-Cell Model Calculations 381
7.2.3.5 Cazacu and Revil-Baudard [16] 3-D Plastic Potentials 383
7.2.4 Porous Materials with Tresca Matrix 400
7.2.4.1 Cazacu et al. [18] Yield Criterion 402
7.2.4.2 Implications of Adopting the Classic Simplifying Hypothesis When Modeling Porous Materials with Tresca Matrix 411
7.2.4.3 Comparison of the Cazacu et al. [18] Yield Criterion with F.E. Unit-Cell Calculations 414
7.2.4.4 Importance of the Local Plastic Heterogeneity on the Dilatational Response of a Porous Tresca Material 419
7.2.4.5 3-D Strain-Rate Potential 424
7.2.4.6 Comparison Between the Theoretical Response of Porous Solids with Tresca and von Mises Matrices 429
7.2.5 Effect of the Relative Weight of the Invariants of the Matrix on Damage Evolution in Porous Materials 437
7.2.5.1 Cazacu and Revil-Baudard [17] Plastic Potential 439
7.2.5.2 Effect of the Matrix Sensitivity to Both Invariants on Yielding 444
7.2.5.3 Influence of the Matrix Sensitivity to Both Invariants on Porosity Evolution 448
7.3 Constitutive Model for Porous Isotropic Metallic Materials with Incompressible Matrix Governed by an Odd Yield Function 454
7.3.1 Cazacu and Stewart [20] Plastic Potential 456
7.3.1.1 Effect of the Matrix Tension–Compression Asymmetry on Yielding 465
7.3.1.2 Influence of the Matrix Tension–Compression Asymmetry on Void Evolution 469
7.3.2 Effect of the Matrix Tension–Compression Asymmetry on Damage in Round Tensile Bars 472
7.3.2.1 Materials with Matrix Characterized by a Constant Strength Differential Ratio 474
7.3.2.2 Materials with Matrix Characterized by an Evolving Tension–Compression Strength Ratio 480
7.3.3 Application to Al: Comparison Between Porous Models Predictions and in Situ X-Ray Tomography Data 486
7.4 Derivation of Plastic Potentials for Porous Isotropic Metallic Materials Containing Cylindrical Voids 493
7.4.1 Statement of the Problem 494
7.4.2 Plastic Potential for a Porous Material with von Mises Matrix 496
7.4.3 Cazacu and Stewart [21] Plastic Potential for Porous Material with Matrix Displaying Tension–Compression Asymmetry 498
7.4.3.1 Exact Solution for the Problem of a Hollow Cylinder Loaded Hydrostatically 499
7.4.3.2 Cazacu and Stewart [21] Strain-Rate Plastic Potential 502
References 510
8 Anisotropic Plastic Potentials for Porous Metallic Materials 513
8.1 Benzerga and Besson [4] Criterion for Orthotropic Porous Materials with Hill [13] Matrix 514
8.2 Stewart and Cazacu [32] Yield Criterion for Orthotropic Porous Materials with Incompressible Matrix Displaying Tension–Compression Asymmetry 523
8.3 Coupled Plasticity-Damage in Hcp-Ti: Comparison Between Stewart and Cazacu [32] Predictions and Ex Situ and In Situ X-Ray Tomography Data 536
8.3.1 Experimental Results in Uniaxial Compression and Uniaxial Tension of Hcp-Ti 537
8.3.2 Yielding of Porous Hcp-Ti 539
8.3.3 Comparison Between Model Predictions and Data 542
8.3.3.1 Comparison Between Predictions of Plastic Deformation and Data on Smooth Specimens 542
8.3.3.2 Comparison Between Model Prediction and XCMT Porosity Measurements for a Smooth RD Specimen 545
8.3.3.3 In Situ XCMT Measurements of Damage Evolution for a Notched RD Specimen of Hcp-Ti and Comparison with Model Predictions 550
8.4 Effects of Anisotropy on Porosity Evolution in Single Crystals Under Multiaxial Creep 558
8.4.1 Creep Models for Porous Single Crystals with Cubic Symmetry 559
8.4.1.1 Plastic Potential for a Porous Crystal with Cubic Symmetry 560
8.4.1.2 Creep Response of Porous Crystals 562
8.4.2 Creep of Fcc Single Crystals 566
8.4.2.1 Effect of the Loading Orientation and Loading Path on the Plastic Response of the Porous Fcc Crystal 567
8.4.2.2 Porosity Evolution for Various Loading Paths and Crystal Orientation 571
8.4.3 Creep of Single Crystals with Tension–Compression Asymmetry 574
8.4.3.1 Effect of Anisotropy and Loading Path on the Plastic Response of Porous Crystals with tension–compression Asymmetry 575
8.4.3.2 Combined Effects of Anisotropy and Tension–Compression Asymmetry on Porosity Evolution 579
References 589