Plasticity-Damage Couplings: From Single Crystal to Polycrystalline Materials

Preface

Contents

1 Mathematical Framework

1.1 Elements of Vector Algebra

1.2 Elements of Tensor Algebra

1.2.1 Second-Order Tensors

1.2.2 Higher-Order Tensors

1.3 Elements of Vector and Tensor Calculus

1.4 Elements of the Theory of Tensor Representation

1.4.1 Symmetry Transformations and Groups

1.4.2 Representation Theorems for Orthotropic Scalar Functions

References

2 Constitutive Equations for Elastic–Plastic Materials

2.1 Stress-Based Formulation of Elastic–Plastic Models

2.1.1 Ideal Plasticity

2.1.2 Elastic–Plastic Work-Hardening Materials

2.1.3 Time Integration Algorithm for Stress-Based Elastic–Plastic Constitutive Models

2.2 Strain-Rate-Based Formulation for Elastic–Plastic Models

2.2.1 Mathematical Framework

2.2.2 Time Integration Algorithm for Strain-Rate-Based Elastic–Plastic Models

References

3 Plastic Deformation of Single Crystals

3.1 Elements of Crystallography

3.2 Plastic Deformation Mechanisms in Crystals: Experimental Evidence

3.2.1 Crystallographic Slip

3.2.2 Deformation Twinning

3.3 Yield Criteria for Single Crystals

3.3.1 Generalized Schmid Yield Criterion

3.3.2 Cazacu et al. [26] Yield Criterion

3.3.2.1 Effect of Loading Orientation on Yielding

3.3.2.2 Procedure for Identification of the Yield Criterion

3.3.3 Application to the Description of Yielding in Cu and Al Single Crystals

3.3.3.1 Cu Single Crystal

3.3.3.2 Al 5% Cu Single Crystal

3.3.4 Application of Cazacu et al. [26] Single Crystal Criterion to Deep Drawing

3.4 Modeling of Plastic Anisotropy of Polycrystalline Textured Sheets Based on Cazacu et al. [26] Single Crystal Criterion

3.4.1 Analytical Expressions for the Yield Stress and Lankford Coefficients of Ideal Texture Components

3.4.1.1 Cube Texture

3.4.1.2 Goss Texture \left\{ {{\bf 110}} \right\} \langle {\bf 001} \rangle

3.4.1.3 Brass Texture \{ \bar{2}1\bar{1}\} \langle 011 \rangle

3.4.1.4 Copper Texture \left\{ {112} \right\} \langle 11\bar{1} \rangle

3.4.1.5 Rotated Cube Texture \left\{ {100} \right\} \langle 011 \rangle

3.4.2 Prediction of Plastic Anisotropy of Textured Polycrystalline Sheets with Several Texture Components

3.4.2.1 Effect of the Spread About Ideal Textures on the Uniaxial Plastic Properties

3.4.2.2 Predictions of Anisotropy of Yield Stresses and Lankford Coefficients for Textured Sheets

3.4.2.3 Applications to Polycrystalline Al and Steel Sheets

References

4 Yield Criteria for Isotropic Polycrystals

4.1 General Mathematical Requirements

4.1.1 General Form of Isotropic Yield Criteria

4.1.2 Representation of the Yield Surface of Isotropic Materials in the Octahedral Plane

4.2 Yield Criteria for Isotropic Metallic Materials Displaying the Same Response in Tension–Compression

4.2.1 Classical Yield Criteria

4.2.1.1 von Mises [44] Yield Criterion

4.2.1.2 Tresca [42] Yield Criterion

4.2.2 Drucker [15] Yield Criterion

4.2.3 Hershey–Hosford Yield Criterion

4.3 Yield Criteria for Isotropic Metallic Materials Showing Asymmetry Between the Response in Tension–Compression

4.3.1 Cazacu and Barlat [8] Yield Criterion

4.3.2 Cazacu et al. [9] Isotropic Yield Criterion

4.4 Application of the Cazacu et al. [9] Yield Criterion to the Description of Plastic Deformation Under Torsion

4.4.1 Monotonic Torsion: Analytical Results

4.4.2 F.E. Simulations of Monotonic Free-End Torsion

4.4.3 Application to Commercially Pure Al

4.5 Cyclic Torsional Loading

References

5 Yield Criteria for Anisotropic Polycrystals

5.1 General Methods for Extending to Anisotropy Yield Criteria for Isotropic Materials

5.1.1 Generalized Orthotropic Invariants

5.1.2 Generalized Transversely Isotropic Invariants

5.2 Orthotropic Generalization of von Mises Isotropic Criterion Due to Hill [22]

5.2.1 Yield Stress Anisotropy Predicted by the Hill [22] Criterion

5.2.2 Variation of the Lankford Coefficients with the Tensile Loading Direction According to Hill [22] Criterion

5.2.3 Comments on the Identification Procedure

5.3 Non-quadratic Three-Dimensional Yield Criteria for Materials with the Same Response in Tension–Compression

5.3.1 Cazacu and Barlat [11] Orthotropic Criterion

5.3.1.1 Predicted Anisotropy in Yield Stresses and Lankford Coefficients

5.3.1.2 Extension of Drucker [16] Isotropic Yield Criterion to Transversely Isotropic Materials

5.3.2 Cazacu [10] Orthotropic Yield Criterion

5.3.2.1 Anisotropy in Lankford Coefficients and Uniaxial Yield Stresses in the Plane (RD, TD)

5.3.2.2 Anisotropy in Yield Stresses in the Other Symmetry Planes

5.3.3 Explicit Expression of the Barlat et al. [4] Orthotropic Yield Criterion in Terms of Stresses

5.3.4 Explicit Expression of the Karafillis and Boyce [28] Orthotropic Yield Criterion in Terms of Stresses

5.3.5 Explicit Expression of Yld 2004-18p Orthotropic Yield Criterion in Terms of Stresses

5.3.6 Explicit Expression of Yld 2004-13p Orthotropic Yield Criterion in Terms of Stresses

5.4 Yield Criteria for Textured Polycrystals with Tension–Compression Asymmetry

5.4.1 Orthotropic Yield Criterion of Cazacu and Barlat [13]

5.4.2 Orthotropic Yield Criterion of Nixon et al. [36]

5.4.2.1 Yielding Formulation

5.4.2.2 Applications: Tension, Compression, and Bending of hcp-Ti

5.4.3 Orthotropic and Asymmetric Yield Criterion of Cazacu et al. [14]

5.4.3.1 Yielding Description

5.4.3.2 Applications: Tension, Compression, and Torsion of hcp-Ti and Mg AZ31

References

6 Strain-Rate-Based Plastic Potentials for Polycrystalline Materials

6.1 Isotropic Strain-Rate Plastic Potentials

6.1.1 Strain-Rate Potentials for Isotropic Metallic Materials with the Same Response in Tension–Compression

6.1.1.1 Exact Duals of the von Mises and Tresca Stress Potentials

6.1.1.2 Hershey–Hosford Pseudo-Strain-Rate Potential

6.1.1.3 Strain-Rate Potential of Cazacu and Revil-Baudard [7]

6.1.2 Strain-Rate Potentials for Isotropic Metallic Materials with Asymmetry Between Tension–Compression

6.1.2.1 Exact Dual of the Isotropic Cazacu et al. [5] Stress Potential

6.1.2.2 Application to Fixed-End Torsion

6.2 Orthotropic Strain-Rate Plastic Potentials

6.2.1 Strain-Rate Potentials for Orthotropic Materials with the Same Response in Tension–Compression

6.2.1.1 Exact Dual of the Hill [14] Stress Potential

6.2.1.2 Orthotropic Strain-Rate Potential of Barlat et al. [2]: SRP93

6.2.1.3 Orthotropic Strain-Rate Potential of Barlat and Chung [4]: SRP2004-18p

6.2.2 Exact Dual of the Orthotropic Cazacu et al. [5] Stress Potential

References

7 Plastic Potentials for Isotropic Porous Materials: Influence of the Particularities of Plastic Deformation on Damage Evolution

7.1 Kinematic Homogenization Framework for Development of Plastic Potentials for Porous Metallic Materials

7.2 Constitutive Models for Porous Isotropic Metallic Materials with Incompressible Matrix Governed by an Even Yield Function

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

7.2.2 Velocity Field Compatible with Uniform Strain-Rate Boundary Conditions

7.2.2.1 Rice and Tracey [58] Velocity Field

7.2.3 Porous Materials with von Mises Matrix

7.2.3.1 Gurson [30] Plastic Potentials

7.2.3.2 Modified Versions of Gurson [30] Criterion

7.2.3.3 Combined Effects of Mean Stress and Third-Invariant on the Mechanical Response According to Cazacu et al. [19] Plastic Potential

7.2.3.4 Void Growth and Collapse According to Cazacu et al. [19] Model and F.E. Unit-Cell Model Calculations

7.2.3.5 Cazacu and Revil-Baudard [16] 3-D Plastic Potentials

7.2.4 Porous Materials with Tresca Matrix

7.2.4.1 Cazacu et al. [18] Yield Criterion

7.2.4.2 Implications of Adopting the Classic Simplifying Hypothesis When Modeling Porous Materials with Tresca Matrix

7.2.4.3 Comparison of the Cazacu et al. [18] Yield Criterion with F.E. Unit-Cell Calculations

7.2.4.4 Importance of the Local Plastic Heterogeneity on the Dilatational Response of a Porous Tresca Material

7.2.4.5 3-D Strain-Rate Potential

7.2.4.6 Comparison Between the Theoretical Response of Porous Solids with Tresca and von Mises Matrices

7.2.5 Effect of the Relative Weight of the Invariants of the Matrix on Damage Evolution in Porous Materials

7.2.5.1 Cazacu and Revil-Baudard [17] Plastic Potential

7.2.5.2 Effect of the Matrix Sensitivity to Both Invariants on Yielding

7.2.5.3 Influence of the Matrix Sensitivity to Both Invariants on Porosity Evolution

7.3 Constitutive Model for Porous Isotropic Metallic Materials with Incompressible Matrix Governed by an Odd Yield Function

7.3.1 Cazacu and Stewart [20] Plastic Potential

7.3.1.1 Effect of the Matrix Tension–Compression Asymmetry on Yielding

7.3.1.2 Influence of the Matrix Tension–Compression Asymmetry on Void Evolution

7.3.2 Effect of the Matrix Tension–Compression Asymmetry on Damage in Round Tensile Bars

7.3.2.1 Materials with Matrix Characterized by a Constant Strength Differential Ratio

7.3.2.2 Materials with Matrix Characterized by an Evolving Tension–Compression Strength Ratio

7.3.3 Application to Al: Comparison Between Porous Models Predictions and in Situ X-Ray Tomography Data

7.4 Derivation of Plastic Potentials for Porous Isotropic Metallic Materials Containing Cylindrical Voids

7.4.1 Statement of the Problem

7.4.2 Plastic Potential for a Porous Material with von Mises Matrix

7.4.3 Cazacu and Stewart [21] Plastic Potential for Porous Material with Matrix Displaying Tension–Compression Asymmetry

7.4.3.1 Exact Solution for the Problem of a Hollow Cylinder Loaded Hydrostatically

7.4.3.2 Cazacu and Stewart [21] Strain-Rate Plastic Potential

References

8 Anisotropic Plastic Potentials for Porous Metallic Materials

8.1 Benzerga and Besson [4] Criterion for Orthotropic Porous Materials with Hill [13] Matrix

8.2 Stewart and Cazacu [32] Yield Criterion for Orthotropic Porous Materials with Incompressible Matrix Displaying Tension–Compression Asymmetry

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

8.3.1 Experimental Results in Uniaxial Compression and Uniaxial Tension of Hcp-Ti

8.3.2 Yielding of Porous Hcp-Ti

8.3.3 Comparison Between Model Predictions and Data

8.3.3.1 Comparison Between Predictions of Plastic Deformation and Data on Smooth Specimens

8.3.3.2 Comparison Between Model Prediction and XCMT Porosity Measurements for a Smooth RD Specimen

8.3.3.3 In Situ XCMT Measurements of Damage Evolution for a Notched RD Specimen of Hcp-Ti and Comparison with Model Predictions

8.4 Effects of Anisotropy on Porosity Evolution in Single Crystals Under Multiaxial Creep

8.4.1 Creep Models for Porous Single Crystals with Cubic Symmetry

8.4.1.1 Plastic Potential for a Porous Crystal with Cubic Symmetry

8.4.1.2 Creep Response of Porous Crystals

8.4.2 Creep of Fcc Single Crystals

8.4.2.1 Effect of the Loading Orientation and Loading Path on the Plastic Response of the Porous Fcc Crystal

8.4.2.2 Porosity Evolution for Various Loading Paths and Crystal Orientation

8.4.3 Creep of Single Crystals with Tension–Compression Asymmetry

8.4.3.1 Effect of Anisotropy and Loading Path on the Plastic Response of Porous Crystals with tension–compression Asymmetry

8.4.3.2 Combined Effects of Anisotropy and Tension–Compression Asymmetry on Porosity Evolution

References