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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
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.
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Seitenzahl: 161
Veröffentlichungsjahr: 2022
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Table of Contents
Cover
Dedication
Title Page
Copyright
Introduction
PART 1: Crystal Elasticity: Dimensionless and Multiscale Representation
1 Macroscopic Elasticity: Conventional Writing
1.1. Generalized Hooke‘s law
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
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
5.3. Elasticity and physical properties
5.4. Influence of porosity and damage on elasticity
5.5. The mystery of the diamond structure
5.6. What about amorphous materials?
5.7. Inelasticity and fine structure of crystals
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
7.3. 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)
10.4. Influence of internal stresses on the vibrations of coated materials
10.5. Application to the determination of internal stresses of coated materials
Conclusion
References
Index
End User License Agreement
List of Tables
Chapter 4
Table 4.1. Discrete data of the elasticity of cubic symmetry (Cij in GPa, Sij in...
Table 4.2. Discrete data of the elasticity of hexagonal symmetry (Cij in GPa, Si...
Table 4.3.
Simplified writing of various approaches for the cubic symmetry
Chapter 5
Table 5.1.
Elasticity–pressure coupling for several covalent materials
Chapter 7
Table 7.1.
Study of the scattering of the elasticity of submicron W films
Chapter 10
Table 10.1.
Values of the internal stresses analyzed by two methods
List of Illustrations
Chapter 2
Figure 2.1. Dimensionless representation of the elasticity of fcc and cc metals....
Figure 2.2. Evolution of elasticity with temperature for fcc and cc metals. For ...
Figure 2.3.
Evolution of the anisotropy of several metals with temperature
Figure 2.4. Correlation between the experimental ratio of moduli and dimensionle...
Figure 2.5. Experimental error on Poisson’s ratio. For a color version of this f...
Figure 2.6. Uncertainty on the dimensionless representation of the elasticity of...
Figure 2.7. Correlation with S11 and S33 of the three other constants: a) –S13, ...
Figure 2.8. Angular representation of the dimensionless elasticity of the hexago...
Figure 2.9. Dimensionless representation of the elasticity of all cubic sub-symm...
Chapter 3
Figure 3.1.
Insertion of atomic springs. For a color version of this figure, see
...
Figure 3.2. Simulation of a traction test on the lattice cell. For a color versi...
Figure 3.3. Spatial anisotropy of the representation of dimensionless elasticity...
Figure 3.4. Dependence of c/a ratio on anisotropy (hexagonal symmetry). For a co...
Chapter 4
Figure 4.1. Simulation of a traction test on polycrystal. For a color version of...
Figure 4.2. Dimensionless representation for the cubic symmetry of the elasticit...
Figure 4.3. Dimensionless representation for the hexagonal symmetry of the elast...
Figure 4.4. Anisotropy determination error related to experimental error measure...
Figure 4.5. Elastic anisotropy of five shades of textured copper alloys. For a c...
Figure 4.6.
Elastic anisotropy of a superalloy obtained by the additive method
Chapter 5
Figure 5.1.
Measurement head and its sample for free tests
Figure 5.2. Experimental setup for the dynamic resonant measurement of elasticit...
Figure 5.3. Single-grained superalloy. The vertical and horizontal directions co...
Figure 5.4. Evolution with temperature of the elastic constants of CSMX-4 supera...
Figure 5.5. Passage from the matrix of the pseudo-monocrystal to that of the tra...
Figure 5.6. Angular elasticity of a superalloy elaborated by directional solidif...
Figure 5.7.
Rafting during a creep test (loading along the vertical axis)
Figure 5.8. Evolution after creep of the elasticity of a single-grained superall...
Figure 5.9. Evolution with temperature of the elasticity of an Inconel 718 after...
Figure 5.10. Evolution of elasticity in time during the precipitation at 680°. a...
Figure 5.11.
TTT diagram of an Inconel 718
Figure 5.12. Evolution of the elasticity of CuAlNi during its transformation. Su...
Figure 5.13. Phase transformation of a titanium alloy. a) Evolution of elasticit...
Figure 5.14. Magneto-elastic coupling of pure nickel (Ben Dhia 2016). For a colo...
Figure 5.15. Amplitude of ΔE effect as a function of the stress level. Adapted f...
Figure 5.16. Evolution of the elasticity of PZT ceramics with temperature (Ben D...
Figure 5.17. Observation of the pores on the surface of a sample of porous silve...
Figure 5.18. Evolution of the Young’s modulus of sintered silver depending on po...
Figure 5.19. Young’s modulus of aluminum of additive manufacturing as a function...
Figure 5.20.
Micro-cracked electrolytic chromium
Figure 5.21. Effect of pressure on the Young’s modulus and damping of silicon (B...
Figure 5.22. Elasticity of SeGe system. a) Rough representation. b) Rationalized...
Figure 5.23. Dimensionless elasticity of amorphous materials. For a color versio...
Figure 5.24. Isothermal damping spectra of a γ’-Ni3Al polycrystal (Gadaud and Ch...
Figure 5.25. Arrhenius diagram for the relaxation of Ni in Ni3Al (Gadaud and Cha...
Figure 5.26. Isothermal damping spectra of YBCO superconductor (Gadaud and Kaya ...
Chapter 6
Figure 6.1. Elasticity as a function of temperature for an isotropic titanium al...
Figure 6.2. Elasticity as a function of temperature for an anisotropic superallo...
Chapter 7
Figure 7.1. Relative variation of the Young’s modulus measured during bending by...
Figure 7.2.
Elasticity of a high-entropy alloy depending on its stoichiometry
Figure 7.3. Comparison of the elasticity of porous silver under massive form or ...
Figure 7.4. Influence of thickness on the apparent elasticity of a SiC coating d...
Figure 7.5.
Superalloy + anticorrosive platinum aluminide + thermal barrier
Figure 7.6. Elasticity with temperature of the superalloy + anticorrosive platin...
Figure 7.7. Evolution of elasticity near the surface of nitrided steel. Adapted ...
Figure 7.8. Elasticity profile perpendicular to a welding by friction. Adapted f...
Chapter 8
Figure 8.1. Comparison of dispersion in the measurement of Poisson’s ratio with ...
Figure 8.2. Evolution of Poisson’s ratio with temperature for various materials ...
Figure 8.3. Evolution with temperature of Young’s and shear moduli for a high-en...
Figure 8.4. Evolution with temperature of Young’s and shear moduli of porous bul...
Chapter 9
Figure 9.1. Vibration mode on a quarter of the plate. For a color version of thi...
Chapter 10
Figure 10.1. Evidence of the relaxation of elaboration stresses for an HIP sinte...
Figure 10.2. Evidence of the relaxation of elaboration stresses for rolled steel...
Figure 10.3. Evidence of the relaxation of elaboration stresses of AlPtNi coatin...
Figure 10.4. Evidence of the relaxation of elaboration stresses of a porous silv...
Guide
Cover
Table of Contents
Title Page
Copyright
Introduction
Begin Reading
Conclusion
References
Index
End User License Agreement
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to Philippe M.†for having conveyed to me histaste for experimental work andhopefully some of his skill
to Jacques W.for having transmitted to me hisscientific curiosity even if itmay involve calling everythinginto 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 Ltd27-37 St George’s RoadLondon SW19 4EUUK
www.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA
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
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.
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 (poly crystals). 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 first-scale 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 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)
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 1Crystal Elasticity: Dimensionless and Multiscale Representation
1Macroscopic 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):
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
