Crystal Elasticity

This book is an original and timeless description of the elasticity of solids, and more particularly of crystals, covering all aspects from theory and elastic constants to experimental moduli. The first part is dedicated to a phenomenological and dimensionless representation of macroscopic crystal elasticity, which allows us to compare all crystals of the same symmetry with the concept of anisotropy and to establish new relations between elastic constants. Multi-scale approaches are then put forward to describe the elasticity at an atomic scale or for polycrystals. The relationship between elasticity and structural or physical properties is illustrated by many experimental data. The second part is entirely devoted to a Lagrangian theory of vibrations and its application to the characterization of elasticity by means of the dynamic resonant method. This unique approach applied to tension-compression, flexural and torsional tests allows for an accurate determination of elastic moduli of structural and functional crystals, varying from bulk to multi-coated materials.

Introduction ix

Part 1. Crystal Elasticity: Dimensionless and Multiscale Representation 1

Chapter 1. Macroscopic Elasticity: Conventional Writing 3

1.1. Generalized Hooke’s law 3

1.1.1. Cubic symmetry 4

1.1.2. Hexagonal symmetry 8

1.2. Theory and experimental precautions 10

Chapter 2. Macroscopic Elasticity: Dimensionless Representation and Simplification 13

2.1. Cubic symmetry: cc and fcc metals 14

2.2. Hexagonal symmetry 19

2.3. Other symmetries 27

2.4. Problem posed by cubic sub-symmetries 27

Chapter 3. Crystal Elasticity: From Monocrystal to Lattice 31

3.1. Discrete representation 31

3.2. Continuous representation for cubic symmetry 34

3.3. Continuous representation for the hexagonal symmetry 37

Chapter 4. Macroscopic Elasticity: From Monocrystal to Polycrystal 43

4.1. Homogenization: several historical approaches and a simplified approach 43

4.2. Choice of “ideal” data sets and comparison of various approaches 47

4.3. Two-phase materials, inverse problem and textured polycrystals 51

4.3.1. Two-phase materials 51

4.3.2. Reverse problem 52

4.3.3. Textured materials 54

Chapter 5. Experimental Macroscopic Elasticity: Relation with Structural Aspects and Physical Properties 57

5.1. A high-performance experimental method 57

5.2. Elasticity of nickel-based superalloys 60

5.2.1. Single-grained superalloy 61

5.2.2. Passage from cubic symmetry to transverse isotropic symmetry 62

5.2.3. Rafting 65

5.2.4. Precipitation in Inconel 718 66

5.3. Elasticity and physical properties 69

5.3.1. Phase transformations 69

5.3.2. Magneto-elasticity 72

5.3.3. Ferroelectricity and phase transformation 74

5.4. Influence of porosity and damage on elasticity 75

5.4.1. Isotropic porosity 75

5.4.2. Anisotropic porosity 77

5.4.3. Micro-cracks and extreme porosity 78

5.5. The mystery of the diamond structure 79

5.6. What about amorphous materials? 81

5.7. Inelasticity and fine structure of crystals 84

5.7.1. Relaxation of substitutional defects 85

5.7.2. Relaxation of interstitial defects 87

Part 2. Lagrangian Theory of Vibrations: Application to the Characterization of Elasticity 91

Introduction to Part 2 93

Chapter 6. Tension–Compression in a Cylindrical Rod 95

6.1. Tension–compression without transverse deformation 95

6.2. Tension–compression with transverse deformation 97

6.3. Determination of E and v of isotropic and anisotropic materials 98

Chapter 7. Beam Bending 101

7.1. Homogeneous beam bending without shear 101

7.2. Homogeneous beam bending with shear 103

7.2.1. Homogeneous beam bending with shear (rotation) 103

7.2.2. Homogeneous beam bending with shear (deformation) 107

7.2.3. Homogeneous beam bending with shear (comparison) 112

7.3. Application to the characterization of the elasticity of bulk materials 113

7.4. Composite beam bending (substrate + coating) 114

7.5. Composite beam bending (substrate + “sandwich” coating) 117

7.6. Application to the characterization of single coatings 117

7.7. Three-layer beam bending 120

7.8. Multi-layered and with gradient in elastic properties of materials 123

Chapter 8. Plate Torsion 127

8.1 Torsion of homogeneous cylinder 127

8.2. Torsion of homogeneous plate 128

8.3. Determination of the shear modulus and Poisson’s ratio for bulk materials 131

8.4. Torsion of composite plate 134

Chapter 9. Thin Plate Bending 137

9.1. Bending vibrations of a homogeneous thin plate 137

9.2. Application to the characterization of thin plate elasticity 139

Chapter 10. Vibration Measurements and Macroscopic Internal Stresses 145

10.1. Experimental evidence of the relaxation of the internal stresses of bulk materials 145

10.2. Internal stresses and homogeneous beam vibration 148

10.3. Analysis of the profile of internal stresses of coated materials (static case) 150

10.3.1. Analysis of the profile of symmetric double coating stresses 150

10.3.2. Analysis of the profile of single coating stresses 151

10.4. Influence of internal stresses on the vibrations of coated materials 154

10.4.1. Influence of internal stresses on the vibrations of coated materials in sandwich configuration 154

10.4.2. Influence of internal stresses on the vibrations of coated materials in single coating configuration 155

10.5. Application to the determination of internal stresses of coated materials 157

Conclusion 163

References 165

Index 171

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