Fundamentals of Strength Principles, Experiment, and Applications of an Internal State Variable Constitutive Formulation

Acknowledgment

Preface

Foreword

Introduction

List of Symbols

1.0 Measuring the Strength of Metals

1.1 How is Strength Measured?

1.2 The Tensile Test

1.3 Stress in a Test Specimen

1.4 Strain in a Test Specimen

1.5 The Elastic Stress Versus Strain Curve

1.6 The Elastic Modulus

1.7 Lateral Strains and Poisson’s Ratio

1.8 Defining Strength

1.9 Stress Strain Curve

Sidebar – Stress State

1.10 The True Stress – True Strain Conversion

1.11 Example Tension Tests

Sidebar – Scalar Stress Representations

1.12 Accounting for Strain Measurement Errors

1.13 Formation of a Neck in a Tensile Specimen

1.14 Strain Rate

1.15 Measuring Strength – Summary

2.0 Structure and Bonding

2.1 Forces and Resulting Energies Associated with an Ionic Bond

2.2 Elastic Straining and the Force versus Separation Diagram

2.3 Crystal Structure

2.4 Plastic Deformation

2.5 Dislocations

Sidebar – Calculating the Magnitude of the Burger’s Vector

2.6 Structure and Bonding – Summary

3.0 Contributions to Strength

3.1 Strength of a Single Crystal

Sidebar – A Primer on Crystal Lattice Planes and Directions Convention

3.2 The Peierls Stress

3.3 The Importance of Available Slip Systems and Geometry of HCP Metals

Sidebar – Computing Areal Density

Sidebar – Computing Planar Spacing

3.4 Contributions from Grain Boundaries

Sidebar – Crystal Plasticity

3.5 Contributions from Impurity Atoms

3.6 Contributions from Stored Dislocations

3.7 Contributions from Precipitates

3.8 Introduction to Strengthening – Summary

Sidebar – A Note on High Temperatures

4.0 Dislocation – Obstacle Interactions

4.1 A Simple Dislocation – Obstacle Profile

4.2 Thermal Energy – Boltzmann’s Equation

4.3 The Implication of 0 K

4.4 Addition of a Second Obstacle to the Slip Plane

4.5 Kinetics

4.6 Analysis of Experimental Data

4.7 Multiple Obstacles

4.8 Kinetics of Hardening

4.9 Summary

5.0 A Constitutive Law for Metal Deformation

5.1 Constitutive Laws in Engineering Design

Sidebar – The Tapered Plate Experiment

5.2 Simple Hardening Models

5.3 State Variables

Sidebar – When does the assumption of path independence lead to unacceptable predictions

5.4 Defining a State Variable in Metal Deformation

5.5 The Mechanical Threshold Stress Model

Sidebar – Comparison with the Zener-Hollomon Equation

5.6 Common Deviations from Model Behavior

5.7 Summary – Introduction to Constitutive Modeling

6.0 Further MTS Model Developments

6.1 Removing the Temperature Dependence of the Shear Modulus

Sidebar – The Shear Modulus

6.2 Introducing a More Descriptive Obstacle Profile

Sidebar – Activation Volume

6.3 Dealing with Multiple Obstacles

Sidebar – The Tapered Plate Experiment (Cont.)

6.4 The Evolution Equation

6.5 Defining the Activation Volume in the Presence of Multiple Obstacle Populations

6.6 Adiabatic Deformation

Sidebar – Temperature Dependence of the Heat Capacity

6.7 Summary – Further MTS Model Developments

7.0 Data Analysis – Deriving MTS Model Parameters

7.1 A Hypothetical Alloy

7.2 Pure Fosium

7.3 Hardening in Pure Fosium

7.4 Yield Stress Kinetics in Unstrained FoLLyalloy

7.5 Hardening in FoLLyalloy

7.6 Evaluating the Stored Dislocation Obstacle Population

7.7 Deriving the Evolution Equation

7.8 The Constitutive Law for FoLLyalloy

7.9 Data Analysis - Summary

8.0 Application to Copper and Nickel

8.1 Pure Copper

8.2 Follansbee and Kocks Experiments

8.3 Temperature Dependent Stress-Strain Curves

Sidebar – Stress-State Dependence of Hardening in Copper

8.4 Eleiche and Campbell Measurements in Torsion

8.5 Analysis of Deformation in Nickel

8.6 Predicted Stress-Stain Curves in Nickel and Comparison with Experiment

8.7 Application to Shock Deformed Nickel

8.8 Deformation in Nickel Plus Carbon Alloys

8.9 Monel 400 – Analysis of Grain Size Dependence

8.10 Copper – Aluminum Alloys

Sidebar – Role of the Stacking Fault Energy

8.11 Summary

9.0 Application to BCC Metals and Alloys

9.1 Pure BCC Metals

9.2 Comparison with Campbell and Ferguson Measurements

9.3 Structure Evolution in BCC Pure Metals and Alloys

9.4 Trends in the Activation Volume for Pure BCC Metals

9.5 Analysis of the Constitutive Behavior of a Fictitious BCC Alloy – UfKonel

Sidebar – Estimating the Variation of with Strain for Adiabatic Tests

9.6 Analysis of the Constitutive Behavior of AISI 1018 Steel

9.7 Analysis of the Constitutive Behavior of Polycrystalline Vanadium

9.8 Deformation Twinning in Vanadium

9.9 A Model for Dynamic Strain Aging in Vanadium

Sidebar – Why all the Scatter?

9.10 Analysis of the Constitutive Behavior of Niobium

9.11 Summary

10.0 Application to HCP Metals and Alloys

10.1 Pure Zinc

10.2 Kinetics of Yield in Pure Cadmium

10.3 Structure Evolution in Pure Cadmium

10.4 Pure Magnesium

10.5 Magnesium Alloy AZ31

10.6 Pure Zirconium

10.7 Structure Evolution in Pure Zirconium

10.8 Analysis of Deformation in Irradiated Zircaloy-2

10.9 Analysis of Deformation Behavior of Polycrystalline Titanium

10.10 Analysis of Deformation Behavior of Titanium Alloy Ti-6Al-4V

10.11 Summary

11.0 Application to Austenitic Stainless Steels

11.1 Variation of Yield Stress with Temperature and Strain Rate in Annealed            Materials

11.2 Nitrogen in Austenitic Stainless Steels

11.3 The Hammond and Sikka Study in 316

11.4 Modeling the Stress-Strain Curve

11.5 Dynamic Strain Aging in Austenitic Stainless Steels

11.6 Application of the Model to Irradiation Damaged Material

11.7 Summary

12.0 Application to the Strength of Heavily Deformed Metals

12.1 Complications Introduced at Large Deformations

12.2 Stress Dependence of the Normalized Activation Energy g_oe

Sidebar – Data Analysis in Presence of a Variable g_oe

12.3 Addition of Stage IV Hardening to the Evolution Law

12.4 Grain Refinement

12.5 Application to Large-Strain ECAP Processing of Copper

12.6 An Alternate Method to Assess ECAP-Induced Strengthening

12.7 A Large-Strain Constitutive Description of Nickel

12.8 Application to Large-Strain ECAP Processing of Nickel

12.9 Application to Large-Strain ECAP Processing of Austenitic Stainless Steel

12.10 Analysis of Fine-Grain Processed Tungsten

12.11 Summary

13.0 Summary of Status of MTS Model Development

13.1 Analyzing the Temperature-Dependent Yield Stress

13.2 Stress Dependence of the Normalized Activation Energy g_oe

13.3 Evolution

13.4 Temperature and Strain-Rate Dependence of Evolution

13.5 The Effects of Deformation Twinning

13.6 The Signature of Dynamic Strain Aging

13.7 Adding Insight to Complex Processing Routes

13.8 Temperature Limits

13.9 Summary

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