Crystal Elasticity [1 ed.] 9781786308221, 1786308223

Table of contents :
Cover
Half-Title Page
Dedication
Title Page
Copyright Page
Introduction
Contents
Part 1. Crystal Elasticity: Dimensionless and Multiscale Representation
1. Macroscopic Elasticity: Conventional Writing
1.1. Generalized Hooke’s law
1.1.1. Cubic symmetry
1.1.2. Hexagonal symmetry
1.2. Theory and experimental precautions
2. Macroscopic Elasticity: Dimensionless Representation and Simplification
2.1. Cubic symmetry: cc and fcc metals
2.2. Hexagonal symmetry
2.3. Other symmetries
2.4. Problem posed by cubic sub-symmetries
3. Crystal Elasticity: From Monocrystal to Lattice
3.1. Discrete representation
3.2. Continuous representation for cubic symmetry
3.3. Continuous representation for the hexagonal symmetry
4. Macroscopic Elasticity: From Monocrystal to Polycrystal
4.1. Homogenization: several historical approaches and a simplified approach
4.2. Choice of “ideal” data sets and comparison of various approaches
4.3. Two-phase materials, inverse problem and textured polycrystals 4.3.1. Two-phase materials
4.3.2. Reverse problem
4.3.3. Textured materials
5. Experimental Macroscopic Elasticity: Relation with Structural Aspects and Physical Properties
5.1. A high-performance experimental method
5.2. Elasticity of nickel-based superalloys
5.2.1. Single-grained superalloy
5.2.2. Passage from cubic symmetry to transverse isotropic
5.2.3. Rafting
5.2.4. Precipitation in Inconel 718
5.3. Elasticity and physical properties
5.3.1. Phase transformations
5.3.2. Magneto-elasticity
5.3.3. Ferroelectricity and phase transformation
5.4. Influence of porosity and damage on elasticity 5.4.1. Isotropic porosity
5.4.2. Anisotropic porosity
5.4.3. Micro-cracks and extreme porosity
5.5. The mystery of the diamond structure
5.6. What about amorphous materials?
5.7. Inelasticity and fine structure of crystals
5.7.1. Relaxation of substitutional defects
5.7.2. Relaxation of interstitial defects
PART 2: Lagrangian Theory of Vibrations: Application to the Characterization of Elasticity
Introduction to Part 2
6. Tension–Compression in a Cylindrical Rod
6.1. Tension–compression without transverse deformation
6.2. Tension–compression with transverse deformation
6.3. Determination of E and v of isotropic and anisotropic materials
7. Beam Bending
7.1. Homogeneous beam bending without shear
7.2. Homogeneous beam bending with shear 7.2.1. Homogeneous beam bending with shear (rotation)
7.2.2. Homogeneous beam bending with shear (deformation)
7.2.3. Homogeneous beam bending with shear (comparison)
7.3. Application to the characterization of the elasticity of bulk materials
7.4. Composite beam bending (substrate + coating)
7.5. Composite beam bending (substrate + “sandwich” coating)
7.6. Application to the characterization of single coatings
7.7. Three-layer beam bending
7.8. Multi-layered and with gradient in elastic properties of materials
8. Plate Torsion
8.1 Torsion of homogeneous cylinder
8.2. Torsion of homogeneous plate
8.3. Determination of the shear modulus and Poisson’s ratio for bulk materials
8.4. Torsion of composite plate
9. Thin Plate Bending
9.1. Bending vibrations of a homogeneous thin plate
9.2. Application to the characterization of thin plate elasticity
10. Vibration Measurements and Macroscopic Internal Stresses
10.1. Experimental evidence of the relaxation of the internal stresses of bulk materials
10.2. Internal stresses and homogeneous beam vibration
10.3. Analysis of the profile of internal stresses of coated materials (static case)
10.3.1. Analysis of the profile of symmetric double coating
10.3.2. Analysis of the profile of single coating stresses
10.4. Influence of internal stresses on the vibrations of coated materials 10.4.1. Influence of internal stresses on the vibrations of coated
10.4.2. Influence of internal stresses on the vibrations of coated
10.5. Application to the determination of internal stresses of coated materials
Conclusion
References
Index
Other titles from iSTE in Materials Science
EULA

Citation preview

Crystal Elasticity

to Philippe M.† for having conveyed to me his taste for experimental work and hopefully some of his skill to Jacques W. for having transmitted to me his scientific curiosity even if it may involve calling everything into question every morning

Series Editor Gilles Pijaudier-Cabot

Crystal Elasticity

Pascal Gadaud

First published 2022 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd 2022 The rights of Pascal Gadaud to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group. Library of Congress Control Number: 2022933477 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-822-1

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Part 1. Crystal Elasticity: Dimensionless and Multiscale Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter 1. Macroscopic Elasticity: Conventional Writing . . . . . . .

3

1.1. Generalized Hooke’s law . . . . . . . . 1.1.1. Cubic symmetry. . . . . . . . . . . 1.1.2. Hexagonal symmetry . . . . . . . . 1.2. Theory and experimental precautions

. . . .

. . . .

. . . .

3 4 8 10

Chapter 2. Macroscopic Elasticity: Dimensionless Representation and Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.1. Cubic symmetry: cc and fcc metals . . . . 2.2. Hexagonal symmetry . . . . . . . . . . . . 2.3. Other symmetries . . . . . . . . . . . . . . 2.4. Problem posed by cubic sub-symmetries

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

14 19 27 27

Chapter 3. Crystal Elasticity: From Monocrystal to Lattice . . . . . .

31

3.1. Discrete representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Continuous representation for cubic symmetry . . . . . . . . . . . . . . . 3.3. Continuous representation for the hexagonal symmetry . . . . . . . . .

31 34 37

vi

Crystal Elasticity

Chapter 4. Macroscopic Elasticity: From Monocrystal to Polycrystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

4.1. Homogenization: several historical approaches and a simplified approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Choice of “ideal” data sets and comparison of various approaches . 4.3. Two-phase materials, inverse problem and textured polycrystals . . 4.3.1. Two-phase materials . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Reverse problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3. Textured materials . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

43 47 51 51 52 54

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

57

. . . . . .

5.1. A high-performance experimental method . . . . . . . . . . . . . . . . 5.2. Elasticity of nickel-based superalloys . . . . . . . . . . . . . . . . . . . 5.2.1. Single-grained superalloy . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. Passage from cubic symmetry to transverse isotropic symmetry . 5.2.3. Rafting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4. Precipitation in Inconel 718 . . . . . . . . . . . . . . . . . . . . . . . 5.3. Elasticity and physical properties . . . . . . . . . . . . . . . . . . . . . . 5.3.1. Phase transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2. Magneto-elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3. Ferroelectricity and phase transformation . . . . . . . . . . . . . . 5.4. Influence of porosity and damage on elasticity . . . . . . . . . . . . . . 5.4.1. Isotropic porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2. Anisotropic porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3. Micro-cracks and extreme porosity . . . . . . . . . . . . . . . . . . 5.5. The mystery of the diamond structure . . . . . . . . . . . . . . . . . . . 5.6. What about amorphous materials? . . . . . . . . . . . . . . . . . . . . . 5.7. Inelasticity and fine structure of crystals . . . . . . . . . . . . . . . . . . 5.7.1. Relaxation of substitutional defects . . . . . . . . . . . . . . . . . . 5.7.2. Relaxation of interstitial defects . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

57 60 61 62 65 66 69 69 72 74 75 75 77 78 79 81 84 85 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 . . . . . . . . . . 6.2. Tension–compression with transverse deformation . . . . . . . . . . . . 6.3. Determination of E and v of isotropic and anisotropic materials . . . .

95 97 98

Contents

Chapter 7. Beam Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Homogeneous beam bending without shear . . . . . . . . . . . . . . . . 7.2. Homogeneous beam bending with shear . . . . . . . . . . . . . . . . . . 7.2.1. Homogeneous beam bending with shear (rotation) . . . . . . . . . 7.2.2. Homogeneous beam bending with shear (deformation) . . . . . . 7.2.3. Homogeneous beam bending with shear (comparison) . . . . . . . 7.3. Application to the characterization of the elasticity of bulk materials 7.4. Composite beam bending (substrate + coating). . . . . . . . . . . . . . 7.5. Composite beam bending (substrate + “sandwich” coating) . . . . . . 7.6. Application to the characterization of single coatings . . . . . . . . . . 7.7. Three-layer beam bending . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8. Multi-layered and with gradient in elastic properties of materials . . .

vii

101

. . . . . . . . . . .

101 103 103 107 112 113 114 117 117 120 123

Chapter 8. Plate Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127

8.1 Torsion of homogeneous cylinder . . . . . . . . . . . . . . 8.2. Torsion of homogeneous plate . . . . . . . . . . . . . . . 8.3. Determination of the shear modulus and Poisson’s ratio for bulk materials . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4. Torsion of composite plate . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

127 128

. . . . . . . . . . . . . . . . . .

131 134

Chapter 9. Thin Plate Bending . . . . . . . . . . . . . . . . . . . . . . . . . .

137

9.1. Bending vibrations of a homogeneous thin plate . . . . . . . . . . . . . . 9.2. Application to the characterization of thin plate elasticity . . . . . . . .

137 139

Chapter 10. Vibration Measurements and Macroscopic Internal Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145

10.1. Experimental evidence of the relaxation of the internal stresses of bulk materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Internal stresses and homogeneous beam vibration. . . . . . . . . 10.3. Analysis of the profile of internal stresses of coated materials (static case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1. Analysis of the profile of symmetric double coating stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2. Analysis of the profile of single coating stresses . . . . . . . . 10.4. Influence of internal stresses on the vibrations of coated materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1. Influence of internal stresses on the vibrations of coated materials in sandwich configuration. . . . . . . . . . . . . .

. . . . . .

145 148

. . .

150

. . . . . .

150 151

. . .

154

. . .

154

viii

Crystal Elasticity

10.4.2. Influence of internal stresses on the vibrations of coated materials in single coating configuration . . . . . . . . . . . . . . . 10.5. Application to the determination of internal stresses of coated materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171

155

Introduction

The first part of this book focuses on the description of crystal elasticity. This field became fully established at the beginning of the 20th century. Its ancestral branches can be referred to as “crystallosapiens” and “elastosapiens”. The ancestor of the first branch can be considered the mineralogist René Just Haüy, the father of geometric crystallography in the 18th century, who observed the features of the cleavage of crystals such as calcite and deduced that they were made up of small orderly stacks, which he named integrant molecules. The concept of atomic lattice dates back to the 19th century, when it was introduced by Gabriel Delafosse, one of Haüy’s disciples. Shortly afterwards, Auguste Bravais formulated the hypothesis of a lattice structure of crystals based on the principles of geometry, according to which he listed 14 different types of crystal lattices that accounted for crystal anisotropy and symmetry properties. Finally, at the beginning of the 20th century, Max von Laue used X-ray diffraction to experimentally confirm Bravais’ works. As for the second branch, its forerunner is beyond any doubt Robert Hooke, who in the 17th century established the linear relation between force and displacement in a loaded spring. Then, the notions of stress and deformation were formulated by Thomas Young at the beginning of the 19th century. Soon afterwards, Augustin-Louis Cauchy established the 3D formalism for the generalized Hooke’s law using the notion of elastic constants and tensor calculus applied to the mechanics of continuous media. As this brief history indicates, by the beginning of the 20th century, the conditions were met for a wide and varied descent.

x

Crystal Elasticity

First of all, Clarence Zener rigorously quantified the elastic anisotropy of cubic symmetry, a concept that will be constantly revisited in this part of the book. For other symmetries, different approaches were also proposed. Experimental characterization also improved. While simple quasi-static tests, such as the tensile test, were previously used to characterize elasticity, dedicated physical methods involving volume or surface elastic waves were used for more precise measurements of macroscopic quantities: ultrasounds, the dynamic resonant method or Brillouin scattering. Due to very strong development, the mechanics of heterogeneous continuous media could transition from the elasticity of the monocrystal and its constants to the elasticity of aggregates (polycrystals). The historical mean-field models elaborated by Woldemar Voigt and Adolph Reuss were the basis for the exploration of numerous approaches. Finally, the recent numerical progresses allowed the optimization of the whole-field methods. Finally, the recent emergence of nanostructures required a zooming in on interatomic interactions in order to describe fine-scale elasticity or to predict the properties of massive materials using dynamic molecular methods. These approaches at various scales, as well as the amount of experimental and theoretical data or data resulting from simulations involving many categories of materials, make it impossible to draw an overall view that describes crystal elasticity in a simple manner. This book proposes a phenomenological approach that addresses this problem: at the macroscopic scale, anisotropy and a dimensionless representation can be used to simplify the mathematical formulation of the classical crystal elasticity, which is reviewed in Chapter 1, and to classify crystal materials according to their anisotropy regardless of their rigidity. This approach is presented in Chapter 2 for cubic and hexagonal symmetries for which the amount of relatively reliable data is quite sufficient. In particular, although dimensionless crystal elasticity only involves the concept of anisotropy, it is still necessary to correctly describe it. At the end of the chapter, the case of sub-structures of cubic symmetry leads to a firstscale transition from the macroscopic level to the nanoscopic level, and therefore to the atomic scale. Based on the analogy of the behavior of a monocrystal at the two scales, Chapter 3 presents this transition that involves

Introduction

xi

the writing of a new Hooke’s law (which we could call nano-Hooke’s law), which deals with spatial rigidity at the atomic level. A second-scale transition is then approached, which passes from monocrystal elasticity to polycrystal elasticity. Chapter 4 is dedicated to this subject. First, the historical cases of mean-field homogenization are approached, and then a much simplified homogenization is proposed, which allows for the phenomenological classification of these more or less various complex approaches found in the literature. It will be noted that anisotropy remains an essential parameter for comparing the elasticity of polycrystals and mechanical approaches. Finally, Chapter 5 is dedicated to illustrating specific cases of crystal elasticity using the data obtained using a specific high-performance vibration method. The description of this method will be followed by multiple examples of the evolution of elasticity in relation to the structural or physical aspects of functional materials.

Robert Hooke’s illustration of the elasticity of metals ( De potentia restitutiva, 1678)

xii

Crystal Elasticity

The second part of this book describes a Lagrangian approach to vibrations. The aim is to propose a unique way to describe simple geometry vibrations. While a numerical calculation is essential for predicting the vibrations of complex structures under widely variable loadings based on the elasticity data of the materials composing these structures, a reverse approach is proposed here: based on the vibrations of structures and the simplest possible loads (plate torsion, beam and plate bending, traction–compression on a cylindrical rod), detected resonance frequencies can be used to retrieve elasticity data on the materials, whether crystalline or not. Four chapters are dedicated to this subject. Furthermore, a high-performance experiment must be set up, as described in the first part, and a proper formalism should be proposed to connect elasticity and vibrations. Various approaches developed in the 20th century can be found in the literature, but this part presents a unique approach that best addresses the problem. It involves writing the Lagrangian of a dynamic system and applying the Hamilton principle to minimize the energy. The case of massive, (multi)coated or gradient materials is presented to address the increasingly diverse current needs. Examples of experimental characterization are presented for various cases as a way to lighten and illustrate the various calculations. Finally, Chapter 10 focuses on the coupling between vibrations and internal macroscopic stresses, and proposes an alternative method for their analysis.

PART 1

Crystal Elasticity: Dimensionless and Multiscale Representation

Crystal Elasticity, First Edition. Pascal Gadaud. © ISTE Ltd 2022. Published by ISTE Ltd and John Wiley & Sons, Inc.

1 Macroscopic Elasticity: Conventional Writing

This chapter reviews the fundaments of classical crystal elasticity. It summarizes the already existing calculations that are scattered throughout the literature with very different notations. The written formalism presented here employs stiffnesses, which are less complex than compliances and better highlight crystal anisotropy. It is also important to note that the transition from theory to experimental applications requires several precautions. 1.1. Generalized Hooke’s law The generalized Hooke’s law gives the linear relations between the components of stress (σij) and deformation (εij) by means of the factors of proportionality, which are the elastic constants (compliance tensor Cijkl or stiffness tensor Sijkl): ε =S

σ or σ = C

ε

[1.1]

This is valid only under the hypothesis of small deformations. This tensor calculus (a fourth-order tensor having a priori 81 independent parameters) is applicable to any anisotropic crystal. Since tensors σij and εkl are symmetric, it can be shown that Cijkl=Cjikl=Cijlk. Moreover, since the tensor results from the double differentiation of interatomic potential energy, it is also true that Cijkl=Cklij. Consequently, the number of independent parameters is limited to

4

Crystal Elasticity

a maximum of 21 for triclinic symmetry crystals, and this is even lower for higher degree of symmetry. 1.1.1. Cubic symmetry Along an arbitrary direction x’ of the crystal, the Cartesian coordinates l, m and n are defined in the orthonormal reference system (x,y,z), which corresponds to the symmetry directions of the crystal of type . They verify the following relation: +

+

=1

[1.2]

For this symmetry, there are only three independent elastic constants (S11, S12 and S44), and the deformations εx, εy, εz, γxy, γxz and γyz are classically defined with respect to the axes of the reference system. Stresses can be written in a very simple form, depending on the applied stress. Consider a traction test along x’ by applying a stress σx: ε = l lε + n

γ

+

m γ 2

γ

n + γ 2

+m

l γ 2

n + mε + γ 2

+ nε

+ [1.3]

Using the elasticity matrix, the tensor calculus yields: ε = S σ + S

σ +σ

[1.4]

ε = S σ + S (σ + σ )

[1.5]

ε = S σ + S (σ + σ )

[1.6]

γ =S τ

[1.7]

Moreover, very simple relations can be obtained as follows: σ =l σ

[1.8]

σ =m σ

[1.9]

σ =n σ

[1.10]

Macroscopic Elasticity: Conventional Writing

5

τ

= lmσ

[1.11]

τ

= lnσ

[1.12]

τ

= mnσ

[1.13]

Young’s modulus along x’ can then be written as: = S (l + m + n ) + (2S

=

+ S )(l m + l n +

n m )

[1.14]

This expression can be rewritten in order to highlight anisotropy: = Σ

=S

− AΣ

[1.15]

= l m + l n + n m function of spatial anisotropy − 2S

A = 2S

[1.16]

− S function of material anisotropy

[1.17]

The function of anisotropy A follows directly from the anisotropy factor shared by all crystals with cubic symmetry, which was introduced by Zener (1948): A

=

(

)

=

[1.18]

Furthermore, the transverse deformation during the same test along the direction y’ of the Cartesian coordinates l’, m’ and n’, perpendicular to x’, can also be written. They verify the following relations: ll + mm + nn = 0

[1.19]

l +m +n

[1.20]

=1

The transverse deformation along this direction can be written as: ε n

= l′ l ε + γ

+

γ

γ

+

+ n′ε

γ

+ m′

γ

+m

+

γ

+ [1.21]

6

Crystal Elasticity

Since the stress components are the same, the following can be written as: ε = S (l l + m m +n n ) + S (lml m + lnl n + mnm′n′)

1 − (l l + m m +n n ) + [1.22]

Given: Σ′

= l l′ + m m′ + n n′

[1.23a]

Equations [1.19] and [1.20] yield: =S

+ Σ′

[1.23b]

The direction y’ was randomly chosen. Consider a third direction z’ perpendicular to x’ and y’ of the coordinates l’’, m’’ and n’’, which verifies the following: l

+m

+n

=1

[1.24]

ll + mm + nn ′ = 0

[1.25]

Defining: =l l

Σ

+ m m +n n

[1.26]

similarly yields: =S

+ Σ′′

[1.27]

The mean transverse deformation can be written as: =

=S

+

(

)

=S

+

[1.28]

It should be noted that only the traction direction is preserved, while the randomly chosen transverse directions disappear. Since in statistical terms

Macroscopic Elasticity: Conventional Writing

7

(infinite medium), all the transverse directions are equiprobable, Poisson’s ratio along an arbitrary direction can be written as: ν

=

=−

[1.29]

A torsion test is now conducted between the directions x’ and y’ to determine the shear modulus by applying τx’y’: γ

/2 = l l′ε +

n

γ

+

γ

γ

+ n′ε

+

γ

+m

γ

+ m′ε +

γ

+ [1.30]

The same relations between εij and σkl (equations [1.4]–[1.7]) are valid, and the new relations that define the stresses are: σ = ll τ

[1.31]

σ = mm τ

[1.32]

σ = nn τ

[1.33]

τ

=

τ

=

τ

=

τ τ

[1.34] [1.35]

τ

[1.36]

Inserting these relations into [1.30] and using [1.24] and [1.25] yield: =S

+ 2AΣ

[1.37]

Similar to transverse deformations, the following relation is obtained along z’ that is orthogonal to x’ and y’: =S

+ 2AΣ

[1.38]

8

Crystal Elasticity

Calculating the half-sum (where directions are equally probable): =S

+ 2AΣ

[1.39]

1.1.2. Hexagonal symmetry For this symmetry, there are now five independent elastic constants (S11, S12, S13, S33 and S44). This symmetry is dealt with exactly the same as the cubic symmetry. For a traction test in the direction x’, relations [1.2] and [1.3] remain unchanged. However, Hooke’s law should be rewritten using this symmetry: ε = S σ + S σ + ε = S σ + S σ +

[1.40] σ

[1.41]

ε = S σ + S (σ + σ )

[1.42]

γ

=S τ

[1.43]

γ

=S τ

[1.44]

γ

= 2(S −

)τ

[1.45]

Young’s modulus along the direction x’ can therefore be written (by considering εx to be identical to equation [1.3]) as follows: =

= (l + m + 2l m )S

+n S

+ n (l + m )(S

+

)

[1.46] Moreover, with relation [1.2], the following expression can be obtained: =

=S

S −2S )

+ n (2S

+S

− 2S ) +

(S

−S

+ [1.47]

It should be noted that S12 is no longer present in the writing of Young’s modulus, but above all, l and m disappear; consequently, the modulus only depends on n, which is the sine of the angle between x’ and z. This reflects a well-known property of the elasticity of hexagonal symmetry, namely its

Macroscopic Elasticity: Conventional Writing

9

transverse anisotropy (conventionally defined in the plane xy; this transverse anisotropy is in fact implicit in equation [1.45]). In order to study Poisson’s ratio, the same approach is taken for cubic symmetry. First, the transverse deformation along the direction y’ can be written as: ε = S (l l + m m + 2lml m ) + S (l l + m m − 2lml m ) + S n n + S (lml m + lnl n + mnm′n′) + S (n (l + m ) + n (l + m ))

[1.48]

The expression in z’ is the same if the index ‘ is replaced by ‘‘. ε = n (−S

= (S −S

+ S ) + n (S

+S

+S

−S

−S

−S )+

+ 2S )

[1.49]

And the resulting expression of Poisson’s ratio depends only on the angle between x’ and z: ν = −

(

= )

(

) (

)

(

)

(

)

[1.50]

Finally, the shear module during the same torsion test defined for cubic symmetry is written as. =

l l +m m +2

+S

+S n n +S (

2lml m −

+

)+

2S (lnl n + mnmn )

[1.51]

Moreover, using the same approach yields: =( 2n (−S

+ +S

−S )+n + 2S

−

2S )

− S

− 4S

+

+

+ [1.52]

10

Crystal Elasticity

The expressions are somewhat complex for this symmetry, especially since while the transverse isotropy is quite visible, anisotropy is well hidden; this point will be revisited in the next chapter. The cases of quadratic and orthorhombic symmetries are beyond the scope of this chapter. As it will be noted in what follows, the amount of reliable data on these symmetries is insufficient. 1.2. Theory and experimental precautions The first problem arises due to the fact that, when applying the formalism to experimental tests, a mathematical indeterminacy occurs. This is illustrated here by traction tests performed on monocrystals with cubic symmetry. Taking the first sample along the direction , equation [1.15] yields: =S

[1.53]

Taking the second sample in the direction of type yields: =S

−

[1.54]

Since three constants need to be determined, measuring the module along the third direction brings no information, as it yields a linear combination of [1.53] and [1.54]. Therefore, another type of measurement should be considered. For example, if a torsion test is conducted along the direction , equation [1.39] yields: =S

[1.55]

The indeterminacy is then lifted; S11 and S44 are determined by the first and third tests. S12 can be deduced from the second test: −

= (

+

−

)

[1.56]

Macroscopic Elasticity: Conventional Writing

11

The measurement precision required to correctly estimate the elastic constants can be easily imagined. The same problems are applicable to the hexagonal symmetry, while a priori two measurement directions are sufficient, but three different types of experiments are necessary. The second difficulty, certainly less known, arises from the strict application of the generalized Hooke’s law. As already noted, for Poisson’s ratio and the shear modulus, the fact of reasoning in an infinite medium (or a sphere) renders all the directions perpendicular to the stress direction equiprobable and simplifies the formalism. However, this formalism cannot be applied for the measurement of a sample of finite dimensions, unless it exclusively involves cylindrical rods. Once again, consider the case of cubic symmetry and analyze Poisson’s ratio along the direction of type . According to equation [1.29], the statistical value is: ν

=−

[1.57]

However, if all these transverse directions are considered individually, this value evolves angularly between two extrema: ν

=−

First extremum

[1.58]

ν

=−

Second extremum

[1.59]

According to the anisotropy, each one corresponds either to a maximum or a minimum. Therefore, for a rectangular section, for example, the weighting calculation of the directions in the formalism is needed, which additionally requires the knowledge about the orientations of the sides of the rectangle. The problem can be solved by measuring Poisson’s ratio or the shear modulus only along the directions of type or in which these two quantities are anisotropic.

12

Crystal Elasticity

As far as the hexagonal symmetry is concerned, the same type of reasoning leads to conducting the measurements of these two quantities only along the direction z in which they are equally isotropic.

2 Macroscopic Elasticity: Dimensionless Representation and Simplification

Crystal elasticity covers a wide range of stiffnesses. For example, for pure metals at ambient temperature, Young’s modulus of a polycrystal varies by about 20 GPa for lead and over 400 GPa for tungsten. Therefore, comparing the elasticity of all these types of crystals may seem impossible, but if it is likely to bring a certain consistency, this can be used to classify all the crystals having the same symmetry. The starting point of this comparison is obviously a dimensionless representation of elasticity. This phenomenological approach can be found in the literature, based on a representation of ratios of elastic constants. A first study (Ledbetter 2007) is cited here, grouping metals with face-centered cubic symmetry, as well as transition metals (actinides and lanthanides) of hexagonal symmetry in a Blackman diagram: C12/C11 versus C44/C11. The various types of crystals are classified on curves parameterized by Zener anisotropy. These studies were then resumed (Paszkiewicz and Wolski 2007) with another type of representation, Every diagrams (Every and Stoddart 1985), using other ratios of constants. This chapter proposes a similar study which optimizes the type of dimensionless representation that can be used to describe orthotropic symmetries in the simplest possible manner.

Crystal Elasticity, First Edition. Pascal Gadaud. © ISTE Ltd 2022. Published by ISTE Ltd and John Wiley & Sons, Inc.

14

Crystal Elasticity

2.1. Cubic symmetry: cc and fcc metals As already noted, due to their high symmetry, cubic monocrystals have only three independent elastic constants: S11, S12 and S44 (or C11, C22 and C44). This symmetry includes sub-symmetries: simple cubic (sc), centered cubic (cc), face-centered cubic (fcc) and the diamond structure (two face-centered cubic structures shifted by a quarter of the large diagonal). As already noted in the first chapter, anisotropy is defined without ambiguity for this symmetry. A slightly different definition is given, considering that it appears more logical to have a zero anisotropic coefficient for an isotropic material: =A

A

−1=

(

)

−1=2

(1 + ν

)−1

[2.1]

E100, G100 and ν100 are, respectively, Young’s modulus, shear modulus and Poisson’s ratio along the symmetry axes (directions of type ). The bulk modulus B obtained by isostatic loading is the only measurable quantity independent of the measurement direction, irrespective of anisotropy: B=

(

)

[2.2]

Before proceeding with the phenomenological description, elasticity data must be compiled. For this symmetry, the reference book of elasticity data was chosen, which is commonly used by physicists and mechanical engineers (Simmons and Wang 1971). As a first step, only data at ambient temperature are considered for cc and fcc metals, for which various authors provide consistent elasticity data sets. This work does not cover the numerous references from this data book, but they are available to the interested reader. These data are given with five significant figures, and it is clear that, as it will be shown further on in this book, the error on these data may be quite considerable. The second step is to choose a dimensionless representation. It seems logical to consider as first parameter the Zener anisotropy ratio ACZ-1. Then, drawing on approaches from the introduction to this chapter, correlations are established between ratios of elastic constants. For the reasons presented in the previous chapter, stiffnesses are going to be used, and for the sake of

Macroscopic Elasticity: Dimensionless Representation and Simplification

15

simplicity, let us start by observing how the S11/S44 and –S12/S44 ratios evolve depending on anisotropy. These results are shown in Figure 2.1. The fcc metals are represented in red and the cc metals in blue. The fcc ones feature a positive anisotropy involving E111 > E110 > E100. Poisson’s ratio ν100= -S12/S11 is positive provided that ACZ-1 ≥ -3/4. The well-known fact that tungsten is anisotropic is confirmed. There is a strong linear correlation (black lines in Figure 2.1); therefore, the following relations can be written as: = + = +



[2.3] [2.4]

Figure 2.1. Dimensionless representation of the elasticity of fcc and cc metals. For a color version of this figure, see www.iste.co.uk/gadaud/elasticity.zip

16

Crystal Elasticity

The removal of anisotropy by a linear combination of equations [2.3] and [2.4] can be used to obtain a new universal relation between elastic constants for the symmetry: S

+ 2S

=

[2.5]

This means that describing symmetry no longer requires three elastic constants, two being sufficient, the third being deduced from equation [2.5]. As for the bulk modulus (defined in [2.2]), the following relation is obtained: B=

(

)

=

[2.6]

While it is well known that B=1/S11=E100 for an anisotropic material, a far more general relation is obtained here, irrespective of anisotropy. This is a highly surprising relation, as it very simply connects S44, pure shear constant, to B, an elasticity quantity that is representative for a triaxial compression.

Figure 2.2. Evolution of elasticity with temperature for fcc and cc metals. For a color version of this figure, see www.iste.co.uk/gadaud/elasticity.zip

Macroscopic Elasticity: Dimensionless Representation and Simplification

17

Let us now analyze the evolution of dimensionless elasticity with temperature. The materials used are the same as those studied in Figure 1.1, and the evolution of their elasticity with temperature is monitored. This is shown in Figure 2.2. In fact, only the minimal and maximal temperatures measured were considered; the ratios of the constants corresponding to these two temperatures were represented and line segments were drawn between the two. It is clear that anisotropy systematically increases with temperature, which simply means that S11 and –S12 increase faster than S44. This evolution can be followed in a more precise manner for three materials with significantly different anisotropies that were measured within the same range of temperature. Figure 1.3 illustrates this evolution. The increase in anisotropy may be expected to become exponential at higher temperatures, upon melting (zero stiffness).

Figure 2.3. Evolution of the anisotropy of several metals with temperature

18

Crystal Elasticity

Dimensionless elasticity can also be represented by ratios of measurable elastic quantities. Figure 2.4 illustrates this with the E110/E100 ratio of Young’s moduli. This example shows an excellent correlation between the values deduced from data in the literature and our approach: =

(

)(

)

(

)(

)

[2.7]

Figure 2.4. Correlation between the experimental ratio of moduli and dimensionless approach. For a color version of this figure, see www.iste.co.uk/gadaud/elasticity.zip

However, the analysis of the correlation between the values deduced from data in the literature and our approach is significantly poorer if Poisson’s ratio is considered along a direction, such that: =

[2.8]

Figure 2.5 shows a wide scattering of the experimental points with respect to the dimensionless approach. The corollary of this scattering is that these elasticity data can be used as a criterion of selection from several data sets in the literature.

Macroscopic Elasticity: Dimensionless Representation and Simplification

19

Figure 2.5. Experimental error on Poisson’s ratio. For a color version of this figure, see www.iste.co.uk/gadaud/elasticity.zip

2.2. Hexagonal symmetry As already noted in the previous chapter, elasticity is described by five independent elastic constants (S11, S12, S13, S33 and S44) for this type of symmetry, which is referred to as transversely isotropic symmetry. The first point involves the definition of anisotropy. Various approaches can be found in the literature. The first approach involves a double anisotropy considering the ratios of Young’s and shear moduli between the longitudinal and transverse directions (Ledbetter 1977), namely, S33/S11 and S44/2(S11-S12), respectively. Other empirical approaches involve either mechanical considerations (Chung and Buessem 1967) or crystallographic considerations. An approach referred to as universal, combining various considerations, was proposed for all the symmetries (Ranganathan and Ostoja-Starzewski 2008). No further details on these approaches are provided here, as their comparison gives a feeling of lack of rigor in the definition of anisotropy.

20

Crystal Elasticity

To make up for this, a new anisotropy is defined by extending Zener’s concept. Considering that the symmetry axes are the longitudinal direction (L) and two orthogonal transverse directions (T), the extension of equation [2.1] yields: A

=2

(

)

(

)

−1

[2.9]

This expression therefore integrates crystallographic aspects and elastic quantities. Resuming equations [1.47], [1.50] and [1.52] with n=1 for the longitudinal direction and n=0 for the transverse directions leads to: A

=

[2.10]

The lack of data on pure metals (Simmons and Wang 1971) is the source of a second difficulty. However, given the availability of lists with comprehensive data on other crystals, particularly on rocks, their reliability may be called into question. Other data (Hearmon 1979) have therefore been compiled. The latter sometimes cross-check the former. Finally, the question of how to choose simple ratios of elastic constants for the phenomenological study of dimensionless elasticity should be addressed. Since there are five constants, the choice is important. Let us start with the ratio S33/S11= ET/EL. Figure 2.6 represents the correlation of this ratio with the previously defined anisotropy. The first point is that besides zinc and cadmium, all the other crystals (all actinides and lanthanides) are grouped over a very narrow range of anisotropy. This should be linked to the fact that most of these crystals are hexagonally close-packed: the c/a ratio of interatomic distances along L and T is about 1.6, while Zn and Cd have a ratio of about 1.9 (Boyer and Gall 1985, pp. 43–48). As a result of testing other definitions of anisotropy, several gaps can be observed, but the indeterminacy is the same. The task of finding a linear correlation, similarly to the cubic symmetry, is burdensome. It can also be noted that data scattering is wider for cubic symmetry, which is certainly due to the experimental difficulties described in section 1.2. Nevertheless, the comparison of their elasticity was studied (Tromans 2011).

Macroscopic Elasticity: Dimensionless Representation and Simplification

21

Figure 2.6. Uncertainty on the dimensionless representation of the elasticity of hexagonal symmetry

Since the dimensionless representation cannot be used, an alternative should be found, based on conventional writing and the newly proposed definition of anisotropy. For this purpose, let us write once again, in a different manner, equations [1.47], [1.50] and [1.52]: =S



+

S

− 2S

−

ν =− =

(

(

(

−S

+ S

−S )

)−



(1 −

)∓( )

+n





(

S



)(S

+S

− 2S

)(

)

)(

−

−S )

[2.12]

)

+

+ 2n (1 −

[2.11]



)(S

+ [2.13]

22

Crystal Elasticity

A color code was used to highlight four linear combinations of constants1: – blue: anisotropy of Young’s modulus between directions L and T; – purple: anisotropy of Poisson’s ratio between directions L and T; – red: anisotropy of the shear modulus between directions L and T; – green: numerator of anisotropy ratio AHZ-1. This writing can be used to establish the four linear functions representative for anisotropy. Let us postulate that they are mutually proportional. Two constants are therefore sufficient to describe the symmetry. In this case, three linear relations are needed to link the other three constants. References are arbitrarily taken as S11 and S33 whose inverses are Young’s moduli measured along L and T. Given the data scattering (Figure 2.6), it can be reasonably thought that measurements are causing the least error. In order to establish three linear relations with S12, S13 and S44, all the possible linear combinations were simulated, and the set of relations verifying the mutual proportionality of the four functions and having the least scattering was selected. Figure 2.7(a)–(c) represents this optimization; there is obviously some scattering, but a tendency towards linearity can be noted. 10 -1

-S (TPa )

Cd

13

8

Zn

6 Mg Pr

Tb

4

Nd

Gd Zr 2

Os Ru Re

Hf

La Sc

Ti

Co

Dy Ho Er Y

0 0

5

10

15

20

25

30

35

-1

S (TPa ) 33

a) 1 For a color version of equations [2.11], [2.12] and [2.13], see www.iste.co.uk/gadaud/ elasticity.zip

Macroscopic Elasticity: Dimensionless Representation and Simplification

80

S

Pr

La

70

Nd

44

Mg

60 Cd

50

Gd Tb

Y 40

Dy

Sc Er

30

Ho

Zr Zn

20 Co

Hf

Ti

10 Ru/Re Os

0 0

10

20

30

40

50

60

70

(9S +S )/4 11

33

b) 12 -S

Pr

12

10

Nd

8

Mg La

6

Y Zr

4 Co 2

Dy Ti Er Ho Sc

Gd Tb

Hf Cd Re Os Ru

0

Zn

-2 -2

0

2

4

6

8

s /2-s /4 11

33

c) Figure 2.7. Correlation with S11 and S33 of the three other constants: a) –S13, b) S44 and c) -S12

23

24

Crystal Elasticity

The following three relations can be written as: ~

[2.14]

−

~

[2.15]

−

~

[2.16]

There remains an uncertainty with respect to the value of the coefficients of linearity, but it is the simplest set of equations optimizing the approach. It is important to note that Poisson’s ratio along L (-S13/S33) is independent of the anisotropy and it is equal to 0.25 whatever the crystal. Let us now resume the various quantities starting with the anisotropy defined by equation [2.10]. Inserting these new relations, the new and highly simplified expression below is obtained: A

−1

[2.17]

Rewriting the other quantities ([2.11], [2.12] and [2.13]) with these simplifying relations yields the following much simpler relations: =1− ν = = +

n (1 − 5n ) (

)

(

)

(−3 + 23n − 20n )

[2.18] [2.19] [2.20]

Based on these new relations, Figure 2.8 gives an angular representation of the dimensionless elasticity between L and T within a range of discretized anisotropy (-0.2, 0, 0.2, 0.4) representative for the range of experimental anisotropy (Figure 2.6).

Macroscopic Elasticity: Dimensionless Representation and Simplification

a)

b)

25

26

Crystal Elasticity

c) Figure 2.8. Angular representation of the dimensionless elasticity of the hexagonal symmetry: a) Young’s modulus, b) Poisson’s ratio and c) shear modulus. For a color version of this figure, see www.iste.co.uk/gadaud/elasticity.zip

Significant variations of Young’s modulus can be noted in the range of experimental anisotropy (Figure 2.8(a)). On the contrary, the shear modulus has small variations (Figure 2.8(c)), but there is a node for the various curves for an angle of 23°, which corresponds to n2=1/3 in equation [2.20]. As for Poisson’s ratio, Figure 2.8(b) shows that νL=νT=1/4, whatever the anisotropy. As a conclusion to this part, although somewhat burdensome, the elasticity of hexagonal symmetry can once again be described with only two constants, from which the other three can be deduced.

Macroscopic Elasticity: Dimensionless Representation and Simplification

27

2.3. Other symmetries Due to the lack of reliable data in the literature, only a theoretical approach of quadratic and orthorhombic symmetries is provided here. Zener anisotropy coefficient can also be extended using the following expression: A

=

∑

( ∑

)

−1

[2.21]

The index c corresponds to the three axes of symmetry of the orthotropic configuration (c=1, 2 or 3). For quadratic symmetries, 1=2. The problem should be addressed similarly to the hexagonal symmetry. Compared to hexagonal symmetries, for quadratic symmetry, there is a sixth elastic constant, S66. Besides the four anisotropy functions, an S66–S44 function should be added, representing the shear anisotropy between axes 1 and 3 or 2 and 3. For the orthorhombic symmetry, three new constants should be added: S22, S33 and S55. Three new functions should therefore be introduced to describe the dimensionless elasticity: S22–S33, anisotropy of Young’s modulus between the directions 1 and 3, S44–S55 and S55–S66 anisotropies of the shear modulus. 2.4. Problem posed by cubic sub-symmetries Getting back to cubic symmetry, section 2.2 focused on the case of pure fcc and cc metals, for which there are perfectly reliable data in the literature. Let us now address the case of other sub-symmetries based on the data in the literature (Simmons and Wang 1971): – simple cubic: mainly metallic oxides and salts (ionic compounds); – fcc and cc: metallic compounds; – diamond structure: exclusively semiconductors, Ge, Si, III–V and II–VI compounds, as well as diamond, obviously. The dimensionless elasticity of all these crystals is shown in Figure 2.9. If scattering slightly increases, the simple formalism proposed for pure metals

28

Crystal Elasticity

(black dashed lines in Figure 2.9) is undoubtedly still valid. There is also a slight increase in the anisotropy range. The diamond structure only exists on a very narrow range of anisotropy, while ionic crystals are present over a much extended range, such as cc metals.

Figure 2.9. Dimensionless representation of the elasticity of all cubic sub-symmetries. For a color version of this figure, see www.iste.co.uk/gadaud/elasticity.zip

Whatever the type of atomic interaction (metallic, ionic or covalent), the dimensionless macroscopic elasticity is identical. This is apparently surprising, since elastic constants resulting from the double differentiation of interatomic potentials vary widely depending on the type of interaction (Interatomic Potentials 2016). But since elasticity is a description of matter

Macroscopic Elasticity: Dimensionless Representation and Simplification

29

around thermodynamic equilibrium, irrespective of the type of potential, the curvature at this point of equilibrium is the only aspect that matters. If elasticity is only dictated by symmetry and anisotropy, this raises the question of the meaning of this result at the atomic scale, which will be addressed in the following chapter.

3 Crystal Elasticity: From Monocrystal to Lattice

When passing from macroscopic to atomic scale elasticity, it must be considered that during a simple test, such as traction, the mechanical behavior must be the same for any monocrystal, and for the lattice cell. Moreover, the description at the nanoscopic scale involves the well-known atomic springs, which represent the interatomic cohesion forces. This type of scale transition can be found in the literature. A first study related to the cubic and hexagonal symmetries addressed this problem by using central force interactions between nearest neighbors (Chen et al. 2015), while another study adding shear and torsion springs focused on isotropic materials (Zhang et al. 2014). This chapter first presents this approach, which allows a first description of the elasticity at the atomic scale by formulating the problem in a simple manner, which will be useful later on. 3.1. Discrete representation First, springs are inserted between nearest neighbors, as shown in Figure 3.1 for sc symmetry (atoms in green). k1, k2, k3 represent the stiffnesses per unit length between nearest neighbors in the directions , and , respectively. For the sake of clarity, the figure shows all the k1 representing the cubic structure, and only one of the other types. When counting them, while apparently there are, respectively, 12, 12 and 8, actually there are 3, 6 and 8, the edges belonging to 4 lattices, and the diagonals belonging to 2.

Crystal Elasticity, First Edition. Pascal Gadaud. © ISTE Ltd 2022. Published by ISTE Ltd and John Wiley & Sons, Inc.

32

Crystal Elasticity

k3

k1

k2

Figure 3.1. Insertion of atomic springs. For a color version of this figure, see www.iste.co.uk/gadaud/elasticity.zip

f001

0

Figure 3.2. Simulation of a traction test on the lattice cell. For a color version of this figure, see www.iste.co.uk/gadaud/elasticity.zip

Crystal Elasticity: From Monocrystal to Lattice

33

Let us now conduct a tensile test along the axis, as illustrated in Figure 3.2 by applying to it a force per unit length f001, while blocking the displacement on the green atom. Denoting by a the lattice constant, the relation between displacements along the symmetry axes and macroscopic deformation is: d = aε

[3.1]

d = −aε

[3.2]

The various displacements between atoms are deduced geometrically (see Figure 3.2). Let us now establish the force balance. Along the direction and along a transverse direction, the following relations are obtained, respectively: =E

=

+ 2k (1 − ν

=0=−

) + k (1 − 2ν

+ 2k (1 − 3ν

)

) + k (1 − 2ν

[3.3] )

[3.4]

A linear combination of these two equations can be used to eliminate Poisson’s ratio and find a general relation similar to [2.6]: k + 4k + 4k =

= 3B

[3.5]

If the same relations are written for cc or fcc symmetries, additional atoms must be considered, respectively, at the midpoint of the longer diagonal and at the center of the lateral faces; hence, the number of springs along certain directions doubles, but since their displacement is halved, the energy balance remains unchanged and the equations are the same. Therefore the atomic stiffnesses kij can be linked to macroscopic elasticity quantities; however, a last linear relation is needed to solve the system of equations. Unfortunately, similarly to the macroscopic tests (see section 1.2), there remains an indeterminacy. This same approach led to a multiplication of the types of tests (biaxial and triaxial tests on the lattice) and the linear combination relations of [3.3] and [3.4] are found. Attempts to simulate tractions in directions for which there is shear have also been done, but the displacement approach is no longer appropriate.

34

Crystal Elasticity

3.2. Continuous representation for cubic symmetry The previous approach has at least the advantage that it presents in a simple manner how to transition between macroscopic and lattice scales. On the contrary, it lacks physical reality. In order to make up for this, a 3D representation of the atomic elastic environment will be attempted; for this purpose, Hooke’s law is rewritten with atomic stiffnesses, forces and displacements. Furthermore, the hypothesis that atomic interactions must obey macroscopic symmetry is also made. This can be formulated by a “nano-Hooke” law: f =k d

[3.6]

where fi represents the force in a (1 ≤ i≤ 3) symmetry direction of the crystal, and dj is the force along the same direction (1 ≤ j ≤ 3); for the symmetry, the stiffnesses per unit length verify the relations k11= k22= k33 and k12=k13=k21=k31=k23=k32. Consequently, there are only two independent elastic parameters, namely, k11 and k12, the first representing an axial stiffness, while the second represents a transverse stiffness. Considering an arbitrary direction of the crystal, the relations in section 3.1 can be rewritten for a traction of axis, but in spherical coordinates (projection of the φ angle on the xy plane and of the θ angle along z) using the following relations:

k

, ,

l = cosθcosφ

[3.7]

m = cosθsinφ

[3.8]

n = sinθ

[3.9]

d

,

k

, ,

= a(sinθ − ν =

cosθ(cosφ + sinφ))

sinθ +

sinθcosθ(cosφ + sinφ)

[3.10] [3.11]

= k cosθ cosφ + k (coθ cosφsinφ + sinθcosθcosφ) [3.12]

The force balance is: =E

=

=

k

, ,

d

,

ds

[3.13]

Crystal Elasticity: From Monocrystal to Lattice

k

=0=

d

, ,

,

ds

35

[3.14]

where ds represents an infinitesimal element of the surface by the eighth of the sphere defined for 0 ≤ φ≤ π/4 and 0≤ θ ≤ π/4. In order to normalize the integration, the following relation can be used: =1=

= ∬ cosθdθdφ

[3.15]

The trigonometric integral calculation is quite lengthy but trivial; therefore, the focus is on the results: = k (1 − ν

) + k (1 − ν

0 = k (1 − 3ν

(1 + ))

) + k ((3 + ) − ν

(3 + ))

[3.16] [3.17]

Resuming the simplifications of section 2.1, this approach can be used to write the nano-/macro-relations: k

=

k

=

(

)

[3.18] [3.19]

First, it should be noted that for an anisotropic material, k12=0. This means that anisotropy corresponds to central cohesion forces. In the other cases, k12 is different from 0, which means that the transverse opposing force can be positive or negative. The spatial dependence of these stiffnesses is expressed as follows: k

=k

,

=k

+ 2(lm + ln + mn)k

[3.20]

With respect to the symmetry, as noted in section 1.1.1, the extrema values of the module were obtained for and directions with the equation [1.15]; their equivalence in stiffnesses can be written as follows: k

=k

k

=k

[3.21] + 2k

=

(

)

[3.22]

36

Crystal Elasticity

Cohesion exists if these quantities are positive, which involves: −0.58 = −

≤A

≤

= 2.67

[3.23]

This yields more or less the experimental range of Figure 2.9. These two points – namely, the logic of the condition of isotropy and the coherence of the range of anisotropy – lead to the validation of this continuous approach. Finally, let us visualize the angular variations of the nano-stiffnesses as a function of anisotropy; this is illustrated by Figure 3.3 for several crystallographic directions. A three-dimensional representation may seem more appropriate; however, having a clear visualization throughout the range of anisotropy would require highly variable angles of representation, given the significant ratio of values of atomic stiffness depending on anisotropy.

Figure 3.3. Spatial anisotropy of the representation of dimensionless elasticity at the atomic scale. For a color version of this figure, see www.iste.co.uk/gadaud/elasticity.zip

Crystal Elasticity: From Monocrystal to Lattice

37

3.3. Continuous representation for the hexagonal symmetry The same approach is applied for this symmetry with the new conditions k11=k22=kT, k13=k31=k23=k32, k12=k21 and k33=kL in order to respect transverse anisotropy. There are already four independent elastic parameters, but since displacements are used, another parameter must be added, namely c/a, the compactness ratio. Furthermore, a difficult problem arises, namely that of the position of atoms in the basal plane for a description with respect to an orthonormal reference system. As already noted, hexagonal symmetry is not simple. To simplify the problem, assume that k12=k21=0 in order to respect the macroscopic transverse anisotropy, as it is the case for anisotropic cubic symmetry. Moreover, consider k13=k31=k23=k32=kLT. And let us rewrite the nano-stiffness as a function of the θ angle between T and L: k(θ) = k cos (θ) + 2k

cos(θ) sin(θ) + k sin (θ)

[3.24]

The simulation of a tensile test along yields: ,

=k

cos(θ) sin(θ) + k sin (θ)

[3.25]

,

=k

cos(θ) sin(θ) + k cos (θ)

[3.26]

“Nano-Hooke” law is applied by defining the displacement along the L direction: d(θ) = csin(θ) − ν acos (θ)(cos(φ) + sin (φ)(1 + cos(60) + sin(60)) [3.27] Compared to the cubic symmetry, it must be taken into account that in the transverse direction, there is an angle of 60° between springs, instead of 90°; therefore, their projection along x- and y-axes should be added. As νL=1/4 (see section 2.2), the displacement along T can be written as follows: d(θ) = csin(θ) − a0.591cos (θ)(cos(φ) + sin(φ))

[3.28]

The force balance is still written as: =

=

k

,

d

,

ds

[3.29]

38

Crystal Elasticity

=0=

k

d

,

,

ds

[3.30]

This yields the first two relations: = 4 (k

cos(θ) sin(θ) + k sin (θ)) sin(θ) cos (θ) −

0.591cos (θ)(cos(φ) + sin(φ)) dθ

[3.31]

cos(θ) sin(θ) + k cos (θ)) sin(θ) cos (θ) −

0 = 4 (k

0.591cos (θ) (cos(φ) + sin(φ)) dθ

[3.32]

After integration, the two relations can be written as: =k k

1 − 0.591

1 − 1.744

+ k (0.785-0.753 )

+ k (0.785-0.753 )=0

[3.33] [3.34]

The same approach is applied with a traction test along . The angular atomic stiffnesses remain unchanged and the displacement is written as: d(θ) = a(cos(θ) cos(φ) − ν acos(θ) sin(φ)) − cν cos(θ)sos(θ)

[3.35]

Moreover, since νT=1/4, two new relations can be written: = 4 (k

= cos(θ) sin(θ) + k cos (θ))

cos (θ)sin(φ) − 4 (k

cos (θ) cos(φ) −

cos(θ)sin(θ) dθ

cos(θ) sin(θ) + k sin (θ))

[3.36]

cos (θ) cos(φ) −

cos (θ)sin(φ) − cos(θ)sin(θ) dθ = 0

[3.37]

Crystal Elasticity: From Monocrystal to Lattice

39

After integration, two new relations are obtained: =k

−

+k (

−

0=k

+k (

− −

)

[3.38]

)

[3.39]

The problem formulation is finally completed; there are four unknowns kL, kLT, kT and c/a, and four linear equations [3.33], [3.34], [3.38] and [3.39]. The solutions of the system are: k =

(4.310 − 1.772 )

k =

.

− 0.958

[3.40]

( − )

k

=

.

(

k

=

.

(1.774 − 1)

[3.41]

− 1)

[3.42] [3.43]

Using the two last equations, KLT can be eliminated to give the relation between the c/a ratio and the elastic constants: .

−1 =

.

(1.774 − 1)

[3.44]

It is a second-order linear equation in c/a that can be rewritten using the definition of anisotropy (equation [2.17]): + (0.297 + 5.391A

) − 3.188 − 9.564A

=0

[3.45]

Let us graphically reconstruct the dependence of c/a on anisotropy and compare it to the data in the literature. All these elements are illustrated in Figure 3.4. The first point is that experimental data (Metal Handbook 1985) are still grouped over a narrow range of anisotropy, with the exception of Zn and Cd. The scattering is still more significant than on the Figure 2.6 as the c/a data, on the one hand, and the elasticity data, on the other hand, are not obtained for the same study materials.

40

Crystal Elasticity

Figure 3.4. Dependence of c/a ratio on anisotropy (hexagonal symmetry). For a color version of this figure, see www.iste.co.uk/gadaud/elasticity.zip

The blue curve represents the results of the approach developed in this section and gives rise to several comments. The first one is that for zero anisotropy, there is a ratio of 1.642, while the compactness gives an ideal ratio of 1.633 (hatched black line). This first point proves at least that the proposed calculation has certain consistency. The ratio increase tendency with anisotropy is also found. The values resulting from solving equation [3.47] are still consistent with the “cloud” of experimental points for negative anisotropies. This is not applicable to Cd and Zn of strongly positive anisotropy. Looking now at the conditions for elastic cohesion, KL and kT must be positive. The lower bound must be reached for 1/S11=0; then equation [3.41] yields c/a=1.5 and AZH-1=-1/3, which seems reasonable. On the contrary, the upper bound must be reached for c/a=2.432 (equation [3.40]), which is inconsistent with the observed experimental values. A further solution is to

Crystal Elasticity: From Monocrystal to Lattice

41

consider 1/S33=0 for the upper bound, which yields a value of 1.774, still too high compared to the experimental range. The proposed approach is therefore not compelling for describing positive anisotropy, which is represented only by two cases. A way to improve the approach would undoubtedly be to consider that k12 is not zero, which means that the initial calculation hypothesis was too strong. This term must be small for low anisotropies, but must become significant for a strong positive anisotropy. On the contrary, this amounts to a fifth parameter to be integrated, and with a still improbable validation given the experimental scattering on the only two cases of strong anisotropy.

4 Macroscopic Elasticity: From Monocrystal to Polycrystal

The passage from the elastic constants of a monocrystal to the polycrystal-characteristic elasticity data involves the transition from the writing of elasticity of a continuous medium to that of a heterogeneous medium composed of randomly oriented monocrystalline grains. This chapter focuses on several historical cases of mean-field homogenization among the multitude of existing approaches by inserting the simplified relations of Chapter 2 for elasticity data sets describing the monocrystal anisotropy. 4.1. Homogenization: several historical approaches and a simplified approach The first to be presented is the Reuss model (1929), which involves a constraint homogenization, and it is applied to a cubic symmetry. Resuming equation [1.15], a random distribution of grain orientation yields the following value: =

Σ

Σ

ds =

[4.1]

For the monocrystal, Σlmn varies from 0 to 1/3. Inserting this value, the following well-known relation is obtained: E

,

=

Crystal Elasticity, First Edition. Pascal Gadaud. © ISTE Ltd 2022. Published by ISTE Ltd and John Wiley & Sons, Inc.

[4.2]

44

Crystal Elasticity

If this writing is rendered dimensionless and the relations between constants established in the second chapter and the Zener anisotropy are added, then: E

(A

,

(

= 0) =

)

=

[4.3]

Similarly, for the shear modulus, based on equation [1.39], the following relation is obtained: G

=

,

=

[4.4]

The same approach is applicable to hexagonal symmetry. Using equations [1.47] and [1.52], the relations between constants established in the second chapter and Zener anisotropy extended to hexagonal structures, and by setting: n2 poly =

cosθsinθ2 dθ =

0

1

[4.5]

3

The following relations are obtained for this symmetry: E

,

S

=

G

,

S

=

)

(

=

[4.6] =

[4.7]

The approach to Voigt model (1887), which is a homogenization of deformation, is similar, the distribution of grain orientation being involved in the same way: E

,

(A

(A

= 0) =

(

= 0) (

)( )(

) )

=

[4.8] G

,

=

=

(

)

[4.9]

Since the bounds represented by Reuss and Voigt approaches are strongly divergent when anisotropy becomes significant, the second historical model (Hashin and Shtrikman (1962a, 1962b)) dealt with cubic symmetry and also

Macroscopic Elasticity: From Monocrystal to Polycrystal

45

proposed two intermediate bounds between Voigt and Reuss bounds. The description of hexagonal symmetry is more complex using this approach (Hashin and Shtrikman (1962a, 1962b)). Self-consistent models were also to a large degree developed using the works of Eshelby on inclusion (Eshelby 1961). A first approach was chosen for the cubic symmetry (Dewit 2008) and a second one for the hexagonal symmetry (Hashin and Shtrikman (1962a, 1962b)). Since the development of these homogenization methods is beyond the scope of this book, it is not presented here. For an isotropic polycrystal (i.e. without crystallographic texture), Reuss and Voigt bounds, Hashin–Shtrikman bounds and the self-consistent estimation were applied as a comparison using the previously described analytical formulae. These formulae were implemented during an internship1 (Schmitt 2019) in the form of Python scripts. During this work, Voigt and Reuss bounds were also evaluated by calculating, respectively, the mean of the rigidity and stiffness tensors of a great number of monocrystals of randomly drawn orientations, previously expressed in the sample base using transfer matrices. A large number of orientations are required for the polycrystal to be considered isotropic and for the thusly obtained bounds to be the closest possible to the theoretical bounds. Finally, a last simplifying analytical approach is proposed. Its underlying principle is that during a tensile test (vertical direction in Figure 4.1), the fact that crystal grains (small cube in Figure 4.1) are in parallel and in series during loading cannot be bypassed. The following relation is valid locally: σ =E ,ε

[4.10]

For the whole crystal, 1≤n≤N and 1≤l≤L. If N and L are large, the statistical distribution of orientations yields: ε =σ ∑ σ =ε ∑

,

,

=

[4.11]

=ε

[4.12]

1 A big thank you to Véra-Lise Schmitt for the work conducted during her Engineering Internship of the 2nd year of ISAE-ENSMA at DPMM-ENDO of Institut P’, and to Loïc Signor, Lecturer at ISAE-ENSMA, for his help.

46

Crystal Elasticity

σM

1

n

N

1 l

monocristal d’orientation quelconque

Monocrystal of arbitrary orientation

L Figure 4.1. Simulation of a traction test on polycrystal. For a color version of this figure, see www.iste.co.uk/gadaud/elasticity.zip

In these equations, the index M represents the macroscopic value. Therefore, on average, at each point of the crystal, the following can be written as: =

.E E

[4.13]

Since by definition it can be written as: E

=

[4.14]

The combination of these last two equations finally yields: E

=

E E

[4.15]

It should be readily admitted that this false homogenization mixing crystallography and Voigt and Reuss approaches will not please mechanical engineers, but it can be used to obtain a new and simple analytical expression of the elasticity of a polycrystal, and as it will be seen further on, it yields perfectly correct results.

Macroscopic Elasticity: From Monocrystal to Polycrystal

47

4.2. Choice of “ideal” data sets and comparison of various approaches The anisotropy was discretized for several values of the two symmetries covering the observed experimental range, and the simplifying relations between constants of Chapter 2 were added to obtain ideal data. Table 4.1 groups these fictitious sets of elasticity for cubic symmetry. The choice was made to “block” S44 or C44 (light blue cells) for the reasons presented in section 2.1. With this representation, it is easy to follow the evolution of Sij or Cij line by line. ACZ-1

-0.5

0

1

2

3

S11

3.33

6.0

11.33

16.67

22

S44

16

16

16

16

16

-S12

0.67

2

4.67

7.33

10

C11

333.3

250.0

208.2

194.5

187.5

C44

62.5

62.5

62.5

62.5

62.5

C12

83.3

125

145.9

152.7

156.2

Table 4.1. Discrete data of the elasticity of cubic symmetry (Cij in GPa, Sij in TPa-1). For a color version of this figure, see www.iste.co.uk/gadaud/elasticity.zip

Rather than presenting the values of elasticity of various approaches in the form of tables, the dimensionless graphical representation can be used to have a very clear comparison (Figure 4.2(a) and (b)). First of all, it should be noted that isotropy is the converging point of all the curves; this proves at least that there was no error in the rationalization of data. A hierarchy of the various approaches clearly exists. The moduli drop with increasing anisotropy. This should be compared to the study of anisotropy evolution with temperature in section 1.2: if the overall stiffness of the constituent monocrystal decreases with anisotropy, the polycrystal should also have elastic properties that drop, whatever the approach. Finally, it should be noted that the proposed simplified approach is not too distant from the self-consistent approach, which is certainly the most reliable.

48

Crystal Elasticity

a)

b) Figure 4.2. Dimensionless representation for the cubic symmetry of the elasticity of a polycrystal: a) Young’s modulus and b) shear modulus (Schmitt 2019). For a color version of this figure, see www.iste.co.uk/gadaud/elasticity.zip

Macroscopic Elasticity: From Monocrystal to Polycrystal

49

The following table groups these fictitious sets of elasticity data for the hexagonal symmetry; once more, a choice was made to “block” S44 or C44 (light blue cells) for the reasons presented in section 2.1. It is important to note the particular values of –S12 for high anisotropy (gray cells). ACZ-1

-0.25

0

0.25

0.5

0.7

S11

6.92

6.4

5.95

5.56

5.22

S33

1.73

6.4

10.42

13.91

17

S44

16

16

16

16

16

-S12

3.03

1.6

0.37

-0.7

-1.63

-S13

0.43

1.6

2.6

3.48

4.24

C11

186.3

187.5

195.9

213.3

244.8

C33

612.1

187.5

125.2

99.5

85.1

C44

62.5

62.5

62.5

62.5

62.5

C12

85.8

62.5

37.8

7.9

-33.6

C13

68.0

62.5

58.4

55.3

52.8

Table 4.2. Discrete data of the elasticity of hexagonal symmetry -1 (Cij in GPa, Sij in TPa ). For a color version of this figure, see www.iste.co.uk/gadaud/elasticity.zip

The dimensionless representation is shown in Figure 4.3. The same consistency of the various approaches can be noted; the simplified approach is still very close to the self-consistent approach. On the contrary, the evolution of modules with anisotropy is completely different from that of cubic symmetry. Due to the very definition of anisotropy in this case, the ratio of stiffness between L and T directions is inverted at isotropy, but overall, the polycrystal integrating all the directions has moduli that increase with the degree of anisotropy, except for the Reuss approach.

50

Crystal Elasticity

a)

b) Figure 4.3. Dimensionless representation for the hexagonal symmetry of the elasticity of a polycrystal: a) Young’s modulus and b) shear modulus (Schmitt 2019). For a color version of this figure, see www.iste.co.uk/gadaud/elasticity.zip

Macroscopic Elasticity: From Monocrystal to Polycrystal

51

As a final note on these very phenomenological comparisons that show the actual consistency in their dimensionless representation, the writing of all these approaches will be simplified. Similar to our simplified approach, the numerical values of all the other approaches will be considered, in an attempt to find potentially existing simple relations with Voigt and Reuss moduli. For the cubic symmetry, these relations exist and are presented in Table 4.3. The difference between numerical values and proposed formulae expressions does not exceed 1% for the most significant anisotropies. Consequently, two simple expressions are the best representation of the elasticity of polycrystals with cubic symmetry, namely, the self-consistent approach and the simplified approach; this amply justifies the introduction of the latter. Concerning the hexagonal symmetry, this simplification is more unlikely. Resuming the same formulae, they remain valid for low anisotropies, but there is a divergence with numerical data for high anisotropies. This may be generated by our relations between elastic constants of section 2.2, which retain a certain uncertainty. But it may also be caused by homogenization approaches that are more difficult to implement.

Increasing Moduli

Approach

Young’s modulus

Shear modulus

Voigt

E

G

HS+

E + 2E 3

G + 2G 3

Self-consistent

0.5(E + E )

0.5(G +G )

Simplified approach

E E

G G

2 HS-

1 1 +E E

2 1 1 +G G

Reuss

E

G

Table 4.3. Simplified writing of various approaches for the cubic symmetry

4.3. Two-phase materials, inverse problem and textured polycrystals 4.3.1. Two-phase materials The simplified approach can be used to readily write the elasticity of a two-phase isotropic polycrystal. Using indices 1 and 2 for the elastic

52

Crystal Elasticity

properties of two-phase components whose volumetric concentration is f and 1-f, equations [4.11] and [4.12] become: ε =σ ( σ = ε (fE

+

)

[4.16]

+ (1 − f)E )

[4.17]

The following relation can be deduced: =

(

)

[4.18]

While precise knowledge on the volumetric concentrations of components is possible, having data on the elasticity of the two components is more difficult, as the two phases are not always under a massive form, which would allow the measurement of their elastic properties. Consider the case of titanium alloys, which are used in aeronautics. There are two large classes of alloys: α and β-metastable alloys largely composed of a hexagonal phase and a cubic phase, respectively. While the hexagonal phase has roughly the same elasticity well known for pure α titanium, obtaining the β phase requires appropriate quenching at high temperature. As the resulting structure is composed of large grains and it is strongly textured, it is very difficult to correctly describe its elasticity (Helstroffer 2018). This case will be illustrated in the next chapter. 4.3.2. Reverse problem For the experimenter who can conduct measurements only on isotropic single-phase polycrystals, retrieving the elastic characteristics of the monocrystal phase is somehow comparable to a reverse problem, whose solving normally requires formulating many hypotheses. With our approach yielding additional relations between Sij and a simple formulation between the elastic characteristics of the mono- and polycrystals, this reverse problem is more easily solved.

Macroscopic Elasticity: From Monocrystal to Polycrystal

For the cubic symmetry, (Epoly/Gpoly=2(1+νpoly) involves: S

=8

−

the

isotropy

of

the

53

polycrystal

=

[4.19]

ACZ-1 can be determined using simple expressions of GR and GV: −1

64

=S G G =

( (

) )(

)

[4.20]

Considering: x=8

−1

[4.21]

3 − 7x + √9 − 2x + 9x

[4.22]

Yields: A

=

Figure 4.4. Anisotropy determination error related to experimental error measurements on the polycrystal

54

Crystal Elasticity

S11 and S12 can then be readily deduced. This is nevertheless quite theoretical, as the precision of Epoly and Gpoly measurements is expected to be quite high if realistic values of the elastic constants and anisotropy are to be found. Figure 4.4 represents the dependency of anisotropy on the experimental parameter x. A point is arbitrarily taken on the curve at x=0.8 corresponding to ACZ-1=0.74. Then, a 5% scattering is considered on Poisson’s ratio, a value of the order of magnitude observed in Figure 2.5. This induces an uncertainty over x of about ±0.1 and therefore an uncertainty of about ±0.6 over the anisotropy ratio. 4.3.3. Textured materials This chapter ends with a discussion of the elasticity of textured polycrystals. While in foundry polycrystals are practically always isotropic, a texture is generated during their forming in view of manufacturing usable parts (rolling, drawing, forging, etc.). The anisotropy of the basic monocrystal leads to an anisotropy of the elastic behavior, which must be integrated. This can be illustrated for the case of a rolled and drawn copper alloy. Five different formings were conducted, and Young’s modulus was measured in the longitudinal direction and in the transverse direction. The measurement method will be described in the next chapter. Figure 4.5 shows these tripled measurements, a different color code being assigned to each shade. The five formings can obviously be classified according to their stiffness. The anisotropy between the transverse and longitudinal directions is signaled by a relative variation of the order of 10% of Young’s modulus. It is also important to note that elasticity variations are far more significant in the longitudinal direction. Experimental tests can also be conducted in samplings whose orientations correspond to loadings of servicing parts, as a means to size them. As an illustration, consider the extreme case of a wall-shaped nickel-based superalloy obtained by wire additive manufacturing. Measurement samples were cut at each 30° between the building direction and the wall transverse directions. The values of its Young’s modulus are shown in Figure 4.6, and a significant and even asymmetric angular dependency can be noted; as a reminder, the values of the monocrystal modulus vary from 130 to 330 GPa, respectively, along the and directions. Here, the values range between 150 and 250 GPa, far from an isotropic polycrystal!

Macroscopic Elasticity: From Monocrystal to Polycrystal

130 Young's modulus (arbitrary units)

transversal measurement

120

longitudinal measurement

110

100

90

80

70

Figure 4.5. Elastic anisotropy of five shades of textured copper alloys. For a color version of this figure, see www.iste.co.uk/gadaud/elasticity.zip

350 GPa building direction 300

250

200

150

100 0

30

60

90

120

150

180

Angle (°)

Figure 4.6. Elastic anisotropy of a superalloy obtained by the additive method

55

56

Crystal Elasticity

It can therefore be concluded that mean-field approaches are not appropriate for an overall description of the elasticity of textured materials, and the amount of data resulting from experiments is limited. Full-field methods should therefore be employed, while keeping in mind the difficulty of integrating in the calculation the texture representation in terms of statistical orientations of the monocrystals.

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

While elasticity was until now approached from a phenomenological perspective, based on ideal monocrystals, things are quite different when actual functional materials are characterized. These may have specific structures or defects at nano- to microscopic scale. Moreover, a coupling between elasticity and certain physical properties is possible. This may sometimes provide surprises to the experimenter. Some cases having sometimes distinctive properties are approached in this chapter. 5.1. A high-performance experimental method As already mentioned in the Introduction, experimental methods that were more physical than the conventional mechanical tests, such as the tension test (Gadaud 2008), were developed at the beginning of the 20th century. Among them is the dynamic resonant method. It derives from the widely used pulse method, which gave rise to an updated international standard (ASTM 2001). Thanks to the progress of instrumentation, the excitation and resonance detection methods were improved at the former LMPM in Poitiers in the 1990s (Mazot et al. 1992). Without getting into further details, an electrostatic measuring bridge controls a capacitor created between the sample and an electrode, which provides contactless measurement. This section only presents three types of experimental tests dedicated to the study of massive crystal materials, and further theoretical developments are presented in the second part of this book. Crystal Elasticity, First Edition. Pascal Gadaud. © ISTE Ltd 2022. Published by ISTE Ltd and John Wiley & Sons, Inc.

58

Crystal Elasticity

From a mathematical perspective, this involves solving the Navier  equation (steady state of an elastic wave of displacement u in a medium in the absence of volume force). The most common tests for the determination of the Young’s modulus are the beam bending tests. There are two experimental methods: free beam bending on wires (the latter are located at the vibration nodes, for example, at 0.224 and 0.776 of the total length of the sample) and clamped bending, the latter being forbidden by the ASTM standard, as clamping introduces random boundary conditions. For a sample of length L, thickness B and density ρ, the free bending frequency FF can be used to determine the longitudinal Young’s modulus with the relation: E = 0.94645. T. ρ.

F

[5.1]

T = T(B/L,ν) represents a shape ratio that tends to 1 when B < < L. Indeed, it takes into consideration the shear effect (Pickett 1945; Spinner et al. 1960). This relation results from the beam theory, according to which the width of the samples is quite small so that the cross section is not deformed. For the tension–compression tests on a rod of length L and diameter d, these two constants can be measured by the use of tension–compression harmonic frequencies FTk (Davies 1938; Spinner and Valore 1958): F

=

( )

/

1−(

)

[5.2]

Other hypotheses for the calculation of vibrations can be found in the literature, but they lead to the same formalism when d