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Mechanics of Solid Interfaces

Mechanics of SolidInterfaces

Edited byMuriel BracciniMichel Dupeux

First published 2012 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, aspermitted 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 theCLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at theundermentioned address:

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The rights of Muriel Braccini & Michel Dupeux to be identified as the author of this work have beenasserted by them in accordance with the Copyright, Designs and Patents Act 1988.

____________________________________________________________________________________Library of Congress Cataloging-in-Publication Data

Mechanics of solid interfaces / edited by Muriel Braccini, Michel Dupeux.p. cm.Includes bibliographical references and index.ISBN 978-1-84821-373-9 (hardback)1. Interfaces (Physical sciences) 2. Fracture mechanics. 3. Solids. I. Braccini, Muriel. II. Dupeux,Michel.QC173.4.I57M43 2012530.4'17--dc23

2012017659

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN: 978-1-84821-373-9

Printed and bound in Great Britain by CPI Group (UK) Ltd., Croydon, Surrey CR0 4YY

Table of Contents

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiMuriel BRACCINI and Michel DUPEUX

PART 1. FUNDAMENTALS . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 1. Interfaces: the Physics, Chemistry andMechanics of Heterogeneous Continua . . . . . . . . . . . . 3Michel DUPEUX and Muriel BRACCINI

1.1. Definition and terminology . . . . . . . . . . . . . . . . . . . 31.2. Energy considerations . . . . . . . . . . . . . . . . . . . . . . 51.3. Elastic behavior of an interface . . . . . . . . . . . . . . . . 81.3.1. Flat interface. . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.2. Effects of elastic coupling. . . . . . . . . . . . . . . . . . 131.3.3. Ellipsoidal elastic inclusion . . . . . . . . . . . . . . . . 15

1.4. Experimental stress analysis techniques . . . . . . . . . . 181.4.1. Digital image correlation . . . . . . . . . . . . . . . . . . 181.4.2. Incremental hole-drilling method . . . . . . . . . . . . 191.4.3. X-ray diffraction. . . . . . . . . . . . . . . . . . . . . . . . 211.4.4. Numerical modeling . . . . . . . . . . . . . . . . . . . . . 23

1.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.6. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Chapter 2. Structure and Defects of CrystallineInterfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Louisette PRIESTER

2.1. What is a crystalline interface? . . . . . . . . . . . . . . . . 27

vi Mechanics of Solid Interfaces

2.2. Definitions and geometric tools todescribe interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2.1. Formation of an interface . . . . . . . . . . . . . . . . . 292.2.2. Coincidence lattice. . . . . . . . . . . . . . . . . . . . . . 322.2.3. Translation lattice of the bicrystal . . . . . . . . . . . 34

2.3. Structure of interfaces: intrinsic dislocations andstructural units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.3.1. Continuous approach to strain at the interface . . . 352.3.2. First discrete approach:Read and Shockley model . . . . . . . . . . . . . . . . . . . . . 362.3.3. Extension of the discrete approach to any grainboundary: Bollmann’s model . . . . . . . . . . . . . . . . . . . 382.3.4. Intrinsic dislocations and atomic-level descriptionof interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.4. Linear interface defects: extrinsic dislocations. . . . . . 462.5. Interaction between dislocations and interfaces:relaxation of interfacial stresses . . . . . . . . . . . . . . . . . . 472.5.1. Slip transmission processes across an interface . . 472.5.2. Relaxation processes in the interface . . . . . . . . . 502.5.3. Interfacial dislocation stress fields . . . . . . . . . . . 532.5.4. Evolution of stress fields over time . . . . . . . . . . . 56

2.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.7. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

PART 2. SINGULARITIES, NOTCHES AND INTERFACIALCRACKS 65

Chapter 3. Singularities and Interfacial Cracks . . . . . . 67Dominique LEGUILLON

3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.2. Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.2.1. A generic case – the V-notch . . . . . . . . . . . . . . . 703.2.2. Calculation of the GSIFs. . . . . . . . . . . . . . . . . . 733.2.3. The case of interfaces: complex singularities . . . . 753.2.4. A particular case . . . . . . . . . . . . . . . . . . . . . . . 76

3.3. Modal mixity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.4. Brittle fracture mechanics . . . . . . . . . . . . . . . . . . . 803.4.1. The Griffith criterion . . . . . . . . . . . . . . . . . . . . 813.4.2. Kinking of a crack out of the interface. . . . . . . . . 83

3.5. Nucleation of cracks . . . . . . . . . . . . . . . . . . . . . . . 853.5.1. Energy condition . . . . . . . . . . . . . . . . . . . . . . . 863.5.2 Stress condition . . . . . . . . . . . . . . . . . . . . . . . . 88

Table of Contents vii

3.5.3. The nucleation criterion . . . . . . . . . . . . . . . . . . 893.6. Deflection of a crack at an interface . . . . . . . . . . . . . 913.6.1. Weak singularity . . . . . . . . . . . . . . . . . . . . . . . 923.6.2. Strong singularity . . . . . . . . . . . . . . . . . . . . . . 94


Chapter 4. Interface Adherence . . . . . . . . . . . . . . . . . . 101Muriel BRACCINI

4.1. Adhesion and adherence. . . . . . . . . . . . . . . . . . . . . 1014.2. Mode mixity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.3. Measurement of adherence . . . . . . . . . . . . . . . . . . . 1074.3.1. Grid method . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.3.2. Pull test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.3.3. Tape peel test . . . . . . . . . . . . . . . . . . . . . . . . . 1094.3.4. Peel test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.3.5. Bulge-and-blister test . . . . . . . . . . . . . . . . . . . . 1134.3.6. Indentation methods (normal and transverse) . . . . 1164.3.7. Wedge test . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.3.8. Four-point bending. . . . . . . . . . . . . . . . . . . . . . 124

4.4. Conclusion: choosing a test . . . . . . . . . . . . . . . . . . . 1264.5. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

PART 3. PRACTICAL APPLICATIONS . . . . . . . . . . . . . . . . . . 135

Chapter 5. Controlling Adherence . . . . . . . . . . . . . . . . 137Thomas PARDOEN, Olivier DEZELLUS and Muriel BRACCINI.

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.2. Multiscale adherence modeling . . . . . . . . . . . . . . . . 1405.3. Nature and control of interface bonds . . . . . . . . . . . . 1455.3.1. Elimination of barriers to adhesion . . . . . . . . . . . 1505.3.2. Modification of interface chemistry . . . . . . . . . . . 1545.3.3. Reactivity and joining . . . . . . . . . . . . . . . . . . . . 1605.3.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

5.4. Dissipative mechanisms . . . . . . . . . . . . . . . . . . . . . 1635.5.The effect of interface geometry . . . . . . . . . . . . . . . . 1735.5.1. Mechanical anchoring . . . . . . . . . . . . . . . . . . . . 1735.5.2. Microtextured interface. . . . . . . . . . . . . . . . . . . 1755.5.3. Biomimetics . . . . . . . . . . . . . . . . . . . . . . . . . . 177


viii Mechanics of Solid Interfaces

Chapter 6. Crack–interface Interaction . . . . . . . . . . . . 189 Eric MARTIN

6.1. Propagation of a crack near an interface . . . . . . . . . . 191 6.2. Criterion of crack deviation by an interface . . . . . . . . 194 6.3. Propagation of an interfacial crack . . . . . . . . . . . . . 202 6.4. Branching criterion of a crack outside an interface . . . 204 6.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6.6. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Chapter 7. Shock Mechanics and Interfaces . . . . . . . . . 211 Michel ARRIGONI, Michel BOUSTIE, Cyril BOLIS, Sophie BARRADAS, Laurent BERTHE and Michel JEANDIN

7.1. Introduction to shock wave mechanics . . . . . . . . . . . 211 7.1.1. Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

7.1.2. Generation of shock waves . . . . . . . . . . . . . . . . . . 213 7.1.3. Shock wave mechanics relationships . . . . . . . . . . 214 7.1.4. Determination of the Hugoniot in plane P–U (the one-dimensional case) . . . . . . . . . . . . . . . . . . . . 220 7.1.5. Passage of a shock between two materials . . . . . . 221

7.2. Damage under shock . . . . . . . . . . . . . . . . . . . . . . . 227 7.2.1. Spallation phenomenon . . . . . . . . . . . . . . . . . . 227 7.2.2. Some damage criteria . . . . . . . . . . . . . . . . . . . . 228

7.3. Application to the shock adhesion test . . . . . . . . . . . 230 7.3.1. Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 7.3.2. Evaluation of the test on Al–Cu samples . . . . . . . 231 7.3.3. Tests on glued assemblages . . . . . . . . . . . . . . . . 236

7.4 Retrospective: recent advances made in shock adherence testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

7.4.1. Technological advances . . . . . . . . . . . . . . . . . . 240 7.4.2. Analytical approaches . . . . . . . . . . . . . . . . . . . 241 7.4.3. Contributions of numerical simulation . . . . . . . . 242

7.5. Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 7.6. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

PART 4. THIN FILMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

Chapter 8. Coating–Substrate Interfaces . . . . . . . . . . . 251 Michel DUPEUX

8.1. Thin films on massive substrates: a typical case . . . . 251 8.2. State of stress in a thin film–substrate specimen . . . . 252

Table of Contents ix

8.2.1. Boundary conditions . . . . . . . . . . . . . . . . . . . . . 2528.2.2. Strain and stress tensors in the film . . . . . . . . . . 2538.2.3. Strain and stress in a planar substrate . . . . . . . . 2558.2.4. Edge effects. . . . . . . . . . . . . . . . . . . . . . . . . . . 259

8.3. Residual strains in thin films . . . . . . . . . . . . . . . . . 2628.3.1. Physical and chemical origin of stresses . . . . . . . . 2628.3.2. Thermoelastic stresses . . . . . . . . . . . . . . . . . . . 2638.3.3. Extrinsic stresses . . . . . . . . . . . . . . . . . . . . . . . 2648.3.4. Intrinsic stresses . . . . . . . . . . . . . . . . . . . . . . . 264

8.4. Determination of stresses in thin films . . . . . . . . . . . 2668.4.1. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2668.4.2. Some frequent tests for the characterizationof thin films and their residual stresses . . . . . . . . . . . . 267

8.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2698.6. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

Chapter 9. Damage in Thin Films on Substrates. . . . . . 273Michel DUPEUX, Muriel BRACCINI and Guillaume PARRY.

9.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2739.1.1. Typical damage . . . . . . . . . . . . . . . . . . . . . . . . 2749.1.2. Elastically stored energy . . . . . . . . . . . . . . . . . . 275

9.2. Layers in tension . . . . . . . . . . . . . . . . . . . . . . . . . 2779.2.1. Typology of damages in layers in tension . . . . . . . 2779.2.2. Energy balance of crack growth at thefilm–substrate interface . . . . . . . . . . . . . . . . . . . . . . 2809.2.3. Stress corrosion cracking . . . . . . . . . . . . . . . . . . 282

9.3. Films in compression . . . . . . . . . . . . . . . . . . . . . . . 2849.3.1. Typology of damage in films under compression . . 2849.3.2. Mechanical modeling of ripples and blisters . . . . . 285


List of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

Foreword

When we observe the objects around us, wherever we look,we must search for a long time to find any that are made upof a homogeneous material, at either the macroscopic ormicroscopic level. The most common steel contains bothferrite and cementite; plastic materials are stiffened bymineral particles enrobed by a polymer matrix; the ceramicused in our daily pots combines crystallites bonded by aglassy phase. Mineral and organic glasses are frequentlystrengthened by a polycarbonate film or coated with an anti-scratch, anti-fouling or photochromic film.

In advanced technology, the search for optimal material/function matching has led to an increasingly frequent use of“multimaterials”, “hybrid materials”, composites, brazed oradhesive bonding, coatings, and structural or functionalmultilayers. Practically all high-performance metallic alloysare strengthened by precipitates of various sizes, whichhinder the dislocation motion that goes along with plasticity,but localize damage and promote crack initiation at the sametime. Microelectronics’ integrated circuits combine fine metaldeposits, insulators and semiconductors of very differenttypes. Depending on the intended application, variousfunctional characteristics are expected of the solid–solidinterfaces that these single block pieces or components

xii Mechanics of Solid Interfaces

contain, but they are all heterogeneous at different scales. Inevery case, an adequate fracture resistance of interfaces isrequired to maintain the principal function and integrity ofthe material.

The question of the mechanical behavior of the interfacesbetween two different materials is thus a critical challenge,as much from an applied perspective as from a fundamentalone. The problems related to its characterization are far frombeing solved, which justifies active and multidisciplinaryresearch in which the necessity of understanding bringsmechanics together with the physics of materials, thechemistry of solids, and the thermodynamics ofmulticomponent equilibria.

We therefore thought it would be useful to compile typicalcontributions towards the approach to this topic in a singlebook. Part 1 (Chapters 1 and 2) presents the basics of themechanical and structural characterization of interfaces.Part 2 (Chapters 3 and 4) is specifically devoted to adescription of the theoretical and experimental tools used toaddress the issue of the initiation and propagation ofinterfacial fractures. Part 3, containing Chapters 5, 6 and 7,illustrates the way in which, given the current state ofknowledge, we address practical problems of interfacialadherence in various geometric and loading conditions, andattempt to handle them. Finally, Part 4 of this book(Chapters 8 and 9) is dedicated to the specific case ofinterfaces between thin films and substrates, which is ofgrowing importance in current practical applications. Eachchapter includes a bibliography that will help readers tofurther their knowledge in this subject.

We make no claim to have exhausted the subject; but havelaid down some milestones: theoretical and experimentaladvances will follow on the heels of all these approaches, aswill other aspects of interface mechanics that can profitablybe associated with a presentation such as the one we have

Foreword xiii

written. Here, we are particularly thinking of the numerical modeling of problems of adherence and interfacial fracture, which surely deserves further development; and of questions of surface and interface elasticity, the importance of which is just beginning to be realized with the advent of nano-objects and nanofilms in research laboratories. These same nano-objects, as part of a matrix or confined by their free surface (paradoxically!) and the interface with their support, present unexpected deformation mechanisms that we are now just discovering. The approaches presented in this book are often illustrated by reference to inorganic materials, the field of activity for most of the contributors to this book; however, the ideas developed are common to all types of systems, and if we tie them in with the profusion of literature about the adherence of organic materials, their similarity becomes apparent. The older and more frequent topic of the mechanical behavior of mobile interfaces with relative sliding has not been addressed as it is well-documented thanks to specialists in wear and tribology.

We could not end this introduction without offering our deep gratitude to those who have helped us, particularly all the contributors who agreed to write a brief presentation on their area of scientific interest. Their names can be found in the various chapters they have written; this book belongs to them as well.

We wish you happy and fruitful reading.

Muriel BRACCINI Michel DUPEUX

June 2012

PART 1

Fundamentals

Chapter 1

Interfaces: the Physics, Chemistryand Mechanics of

Heterogeneous Continua

Of what does an interface between two solids consist?What qualifying or quantitative physical or chemicalparameters must we specify to define it? What generalconsiderations can we set out about the mechanical behaviorof an interface in a heterogeneous solid? These are the basicquestions that this chapter proposes to answer as anintroduction to the following chapters, focusing on morespecific points.

1.1. Definition and terminology

Strictly speaking, an interface can be defined as the two-dimensional border area between two dissimilar materials.These two materials may differ in their physical state (suchas in the case of solid–liquid or solid–gas interfaces); theirchemical composition (such as an interface between twoimmiscible liquids in an emulsion); their structure (such as a

Chapter written by Michel DUPEUX ANDMuriel BRACCINI.

4 Mechanics of Solid Interfaces

residual martensite/austenite interface in quenched steel);their relative orientation (such as the twin boundaries orgrain boundaries in a polycrystal or the interface betweenlayers in a multilayer composite); or even by their relativetranslation (such as a stacking fault surface in a crystal). Inthis book, as the title indicates, we will focus on the case ofinterfaces between two solid materials.

The geometric aspect of a solid–solid interface can proveextremely variable, depending on the scale at which it isobserved. Abrupt interfaces, in which the physical andchemical characteristics change suddenly as the interface iscrossed (see Figure 1.1a), are an ideal and simple case that israrely encountered in reality. In real cases, the physical orchemical interaction between the two materials disturbstheir composition or their structure in a layer of varyingthickness near the interface, producing what we call adiffuse interface, with or without a marked discontinuity ofproperties (see Figures 1.1b and 1.1c).

Composition

Distance

a) b)

Composition

Distance

A B

c)

Composition

Distance

A B

d)

A B

Composition

Distance

e)

A B

C

Composition

Distance

A B

20 µm

AM1 NiAlZr

f)

Figure 1.1. View and development profile of the physicochemicalcharacteristics (such as chemical composition) across various types ofinterfaces: a) abrupt interface; b) continuous diffuse interface; c) diffuse

interface with discontinuity; d) heterogeneous diffuse interface; e) interfacewith interphase; and f) real interface between an AM1 nickel-basedsuperalloy and a NiAlZr coating (cross-section, scanning electron

microscope (SEM)) [THE 07]

Interfaces: Heterogeneous Continua 5

This heterogeneity may even appear as a complexinterpenetration of particles or protrusions of adjacentmaterials (see Figure 1.1d). Finally, the interface may haveone or more intermediate layers, or interphases, composed ofmaterial that is different from the join between two bulkmaterials that have either been inserted voluntarily duringthe development of a synthesis interface (for example, alayer of adhesive in a glued interface) or are the result of asolid state reaction between the two basic materials (seeFigure 1.1e and Chapter 5).

Even without taking a closer look at the crystalline oratomic scale (see Chapter 2), it is clear that a real interfacealmost always has some “thickness”; the ideal notion of aninterface reduced to a surface of abrupt separation betweentwo solid media, such as the one in Figure 1.1a, is in realityonly a convenient, simplified assumption that is frequentlyused, keeping in mind that it is only valid on a scale ofsufficient width.

1.2. Energy considerations

On the free surface of a solid, or along an internalinterface, the interaction stresses acting on an atom ormolecule due to its bonds with its neighbors are obviouslydifferent from the ones in the bulk of the same solid; sincethe local atomic environment is neither homogeneous norisotropic, the result of these interactions is not zero andleads, for example, to atomic reconstructions on the surfaceor to interatomic distances with equilibrium values that aredifferent from those in the bulk of the material. The same istrue for electronic distributions. In the case of the freesurface of a homogeneous solid, the result is an increase ininternal energy γS in the free surface area (in J/m²) incomparison with any surface embedded within the volume ofthe solid. This surface energy γS includes an enthalpycontribution due to the local chemical composition, which is


different from that within the solid, and an entropycontribution related to the difference in atomic structurebetween the surface and the volume of the solid. In the caseof an interface between two media A and B, for similarreasons each unit of area of the interface representsadditional internal energy γI, the value of which can beexpressed by the Dupré relation (see Chapter 5):

γI = γA + γB – γAB. [1.1]

where γA and γB are the energies of the surfaces of materialsA and B when they are free and γAB represents the energy ofthe interatomic bonds per unit of interface area establishedbetween the two materials at the moment when the interfaceis produced. This energy is usually negative (with the usualconventions of thermodynamics); that is, the side-by-sideplacement of the two free surfaces leads to a reduction ininternal energy, and the interface thus created is more stablethan the two separate elements. Its value, on the basis ofboth theoretical considerations and experimental results, intypically on the order of 1 J/m².

Returning to the case of the total free area of a solid,where the formulation of a physical demonstration issimpler, two different types of variations can be imagined forits total energy, WS:– with the extent of area A of the free surface of interest,at constant interatomic distances (the strain ε beingconstant) we have:

. .S S Scst cstd W d A dAε εγ γ= =⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦= = [1.2]

which leads to the definition of the surface energy γSmentioned above;– with the extent of area A, but at a constant number ofatomic sites, N, for variable strains εij (i, j = 1, 2) in the plane(x1, x2) of the surface, we have:


. . .==⎡ ⎤ ⎡ ⎤⎣ ⎦⎣ ⎦ = = ij ijS N cstN cstd W d A A f dγ ε [1.3]

which leads to the definition of a second-order, two-dimensional tensor fij, the elements of which have thedimension of stresses, called surface stresses. Writing thedifferential of the previous total energy (equation [1.3]) asd(γ.A) = γ.dA + A.dγ and expressing the variation of area A(at a constant number of sites) as dA = A.δij.εij (i, j = 1, 2, andδij being the Kronecker symbol), we get the expression of thesurface stresses in relation to the surface energy γS and itsderivative with regard to the strains of the surface plane:

. Sij ijS

ijf γγ δ

ε∂= +∂

[1.4]

The variations dεij of the surface strains are, in this case,the ones imposed by the interior volume of the solid on all ofthe surface atoms, in comparison to the state of equilibriumthat this two-dimensional layer of atoms could exhibit if itexisted alone in a hypothetical “neutral” reference state. Thisleads to the idea that there are solid free surfaces that, evenif no external stress is applied, are spontaneously in biaxialtension, and others are in biaxial compression in comparisonto the underlying volume. The same deduction applied to aninterface leads us to consider the existence of interfacestresses resulting from similar physical effects. The presenceof such surface–interface stresses leads to instability of theplanarity of the surfaces concerned (the Grinfeld instability);their topography evolves at equilibrium toward a periodicwavy geometry resulting in a relaxation of the total elasticenergy stored. The period and amplitude of the undulations,which are experimentally detectable, depend on the intensityof the surface–interface and volume stresses [GRI 94, BOS99].


1.3. Elastic behavior of an interface

1.3.1. Flat interface

In order to establish a basis to address the problem of theelastic behavior of a dual material containing an interface,let us first imagine the case of a flat abrupt interfacebetween two materials, A and B, that are semi-infinite andsupposed to be elastic, isotropic and homogeneous. Thisensemble will be attached to a system of orthonormal axesOx1, x2 and x3 (see Figure 1.2).

Ox1x2

x3

Material A

Material B

M dS

Figure 1.2. Reference orthonormal frame for a flat interface (x1, x2)between two semi-infinite materials, A and B

In each of the two materials – A (where x3 > 0) and B(where x3 < 0) – the displacement fields will be called UA andUB and the stress and strain tensors will be called [ ]BorA

ijσand [ ]BorA

ijε , respectively. In each material, these tensors are

linked by Young’s equations, which bring in the Young’smodulus EA or B and the Poisson coefficient νA or B; or by theiropposites, Lamé equations, which introduce the Lamécoefficients λA or B and μA or B of the material. At the interface(in the plane x3 = 0), what are the relationships we can detectbetween the fields prevailing in material A and thoseprevailing in material B? To decide this, it will be useful touse the specific limit condition assumptions governing theproperties of this interface.


1.3.1.1. Detached interface

Here, we consider a case where, even with the twomaterials being unable to interpenetrate, there is nointerfacial connection force resisting their separation orrelative displacement. This assumption is only of interest ifit applies to part of the interface of interest: for example, onany half-plane x1 ≥ 0 in Figure 1.2. This is the case of aninterfacial crack where, depending on the external loadapplied and the mechanical characteristics of materials Aand B, the displacement and stress fields take a form thatwill be the subject of a specific description in Chapter 3.

1.3.1.2.Mobile interface

Now let us consider a case where, while preservingcontact along the length of the interface, a relative slipdisplacement of the two adjacent materials is possible; forexample, in direction x1 of the interfacial plane. Thedisplacement fields of the two materials must then satisfythe following conditions, where x3 = 0, for any (x1, x2):

BA UU 11 ≠ , BA UU 22 = and BA UU 33 = [1.5]

All of the partial derivatives of these functions withregards to variables x1 and x3 are a priori different inmaterials A and B, but identical in relation to x2, thedirection of the interface perpendicular to the relativehypothetic displacement. For strains this causes, wherex3 = 0, for any (x1, x2):

A Bi i

BAjj

U Uxx

∂ ∂≠∂∂

for any (i, j), except 2 2

22

A B

BAU U

xx∂ ∂

=∂∂

, hence

22 22A Bε ε= [1.6]

In terms of stresses, all of the components applied on theinterfacial facets (normal to x3) are entirely transmitted from


one material to the other: where x3 = 0, for any (x1, x2),Bi

Ai 33 σσ = . However, the relative displacement in direction x1

limits the value of component σ13, which depends on the lawof friction chosen between the two materials. Thus, forCoulombian-type solid friction, this component will belimited to a threshold value σ13 = τ, below which no relativemovement exists, and above which an indefinite relativedisplacement is possible. In another hypothesis, for viscousfriction: 131 1 .A BU U t ησ∂ − ∂ = . The developments of these

types of contact laws belong to the field of tribology [FEL 03].Naturally, any relative displacement of the facets on whichthe stress component σ13 is applied involves some energydissipation.

1.3.1.3. Perfectly adherent interface

This interfacial condition assumes perfect identification ofdisplacements and the perfect transmission of stresses alongthe interface. The equal values of displacements at theinterface 1 2 1 2( , ,0) ( , ,0)A B

i iU x x U x x= implies equality of thepartial derivatives of these functions with regards tocoordinates x1 and x2; and therefore, for x3 = 0, the similarityof some components of the strain tensors:

1311 12

2321 22

31 32 33

A B A B

Aor B A B A Bij

ε ε ε

ε ε ε εε ε ε

≡ ≡

≡ ≡

⎡ ⎤⎢ ⎥⎢ ⎥⎡ ⎤

⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

= [1.7]

Similarly, the transmission of stresses through theinterface implies equal values of all of the stress componentspossessing index 3 between materials A and B where x3 = 0,for any (x1, x2):


11 12 13

21 22 23

31 32 33

A B

Aor B A Bij

A B A B A B

σ σ σ

σ σ σ σ

σ σ σ

≡

≡

≡ ≡ ≡

⎡ ⎤⎢ ⎥⎢ ⎥⎡ ⎤

⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

= [1.8]

If materials A and B behave in a linear elastic manner,the components of the two strain and stress tensors arelinked by the six Young’s equations (parameters EA or B andνA or B) or their opposites, the six Lamé equations (parametersλA or B and μA or B) in each of the two media. We can then seethat if the values of the six components common to the mediaA and B at the interface are known (three strain and threestress components), we address a well-defined, mixedproblem and must be able to calculate the values of the sixstrain and stress components that are still unknown alongthe interface using linear elasticity equations. Theintegration of strains must then allow us to work out thedisplacement field along the interface, and then in the wholeof the dual-material solid using the appropriate boundaryconditions.

1.3.1.4. Example: compression of a dual-material bar in arigid channel

Take a dual-material bar compressed parallel to the flatinterface by two rigid plateaus, and held laterally betweenthe two rigid walls of a channel with a constant width (seeFigure 1.3). The sample is referred to the orthonormal framedefined in Figure 1.2, the three axes of which in this case arethe principal stress and strain axes, with direction x1 beingthe direction of compression. Its initial dimensions are 2l1,2l2 and 2l3 respectively, in the directions of the threecoordinate axes. For a given relative compression 111 11

A B eε ε= =(< 0), in this case the six components common to the tensorsin materials A and B are known: 12 22 0A B A Bε ε≡ ≡= = (imposed


constant width), and 13 23 33 0A B A B A Bσ σ σ≡ ≡ ≡= = = (free surfaces

normal to x3). The elasticity equations lead to:

1

1

0 00 0 0

0 0(1 )

Aor Bij

Aor B

Aor B

e

eε

νν

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

=

−−

( )

( )

12

12

0 01

0 01

0 0 0

Aor B

Aor B

Aor B Aor BAor Bij

Aor B

E e

E e

ν

νσν

⎡ ⎤⎢ ⎥⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥

⎡ ⎤ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥

⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

−

=−

[1.9]

Rigid sides

Rigidplateau

A

B

x1

x2x3

O

Figure 1.3. Parallelepiped dual-material sample compressed parallel tointerface in a channel with rigid walls (partial

representation of side x1 > 0)


The component functions of the displacement field in thetwo materials are straightforwardly obtained through directintegration, supposing that the origin of the axes is fixed:

1 11Aor BU e x= , 2 0Aor BU = , 1

33 (1 )

Aor BAor B

Aor BeU xνν

−=−

[1.10]

In reality, it is rare for a mechanical problem to have suchsimple and uniform solutions in the dual-material specimenof interest. In most real situations, the boundary conditionsat the interface cause the unavoidable production of localdisturbances, as illustrated by the example below.

1.3.2. Effects of elastic coupling

Imagine we have a very long prismatic elastic dual-material bar of constant width loaded in an apparentlysimple state of traction via uniform displacement imposed atits ends, perpendicular to the transversal plane of itsinterface, which is supposed to be perfectly adherent (seeFigure 1.4a).

x3

A

x2

B

x1

O O

a) b) c)

A33σ

AB3333 σσ =

O

BorABorA1323 σσ =

Figure 1.4. Prismatic dual-material bar in traction perpendicularto the interface: a) configuration at rest; b) deformed configurationschema with tensile stresses at a long distance from the interface

and perturbations due to the compatibility of the transversal strainsalong the interface; and c) peaks of shear stress σ23 and σ13 and

a singular line at the perimeter of the interface


Before even attempting any calculation, we mustacknowledge that the transversal contraction of each part ofthe bar is different a priori, due to the two different values ofPoisson ratios νA and νB. The mutual adaptation of thestrains of the two materials along the adherent interfacerequires the presence of additional strains in comparison to ahypothetical situation in which part A or B freely deforms(see Figure 1.4b); these are accompanied by unexpectedshear stresses 31

Aor Bσ and 32Aor Bσ , which are zero at the

center of the bar and of increasing intensity toward theperiphery of the interface (see Figure 1.4c). The conditions ofequilibrium in material A or B mean that variations alsoappear locally for the other stress components, which areconstant or zero at a long distance from the interface. Alongthe length of the perimeter surrounding the interface, theshear stresses 32

Aor Bσ and 31Aor Bσ would be maximum except

for the fact that, by reciprocity, the presence of the freesurface causes stresses 23

Aor Bσ and 13Aor Bσ to vanish along

this line. This inconsistency of the intuitive values of thestresses and their sudden variations shows the presence of asingular line along the length of the contour of the interface,a line along which the values of some stress componentsdiverge to infinity, at least in theory, as shown by Bogy[BOG 68]. The theoretical justifications of this singularityand its formal description within the context of linearelasticity will be the subject of a specific discussion inChapter 3. To summarize, we will simply say that:

– as is often the case, the compatibility of the elasticstrains of two adjacent different materials causes additionalstrains and stresses near the interface. This creates complex,non-homogeneous fields that need to be determined anddescribed;

– when the interface is of finite dimensions, itsintersection lines with free surfaces usually constitute


singular lines, with spots of stress concentration requiring aspecific approach.

The intensity and complexity of such effects of elasticcoupling, which constitute the most frequent situation,depend on the geometry of the dual-material sample, on theapplied boundary conditions and on the heterogeneity of theelastic properties of the pair of materials. More precisely,Dundurs showed [DUN 69] that within the theory ofisotropic elasticity, the parameters of use in quantifying theelastic contrast of a pair of materials are two combinations, αand β, of the elasticity constants of materials A and B (seeequations [6.2] and [6.3] in Chapter 6), which have sincebeen known as Dundurs coefficients. Another parameter, ε(defined by formula [3.6] in Chapter 3), is convenientlyintroduced to study the effects of elastic coupling. Thesethree parameters are all zero in the case of similarity of theelastic constants of materials A and B and, depending on thesign and intensity of the contrast, vary between the followinglimits (see Chapter 6):

1 1α− ≤ ≤ + , ( 1) ( 1)4 4

α αβ− +≤ ≤ , 0.175 0.175ε− ≤ ≤ + [1.11]

It is therefore all the more necessary to pay attention tothese elastic coupling effects, since they are undoubtedly thecause of localized damage that may lead to the ruin of dual-material joints.

1.3.3. Ellipsoidal elastic inclusion

Among the rare situations in which an analyticalapproach can be used for a dual-material solid is that of aspherical or ellipsoidal elastic inclusion in an infinite matrix,studied by Eshelby [ESH 57, ESH 59, ESH 61]. There aremany practical applications of this on the scale of themicrostructure of materials, from the reinforcement of

mechanicsofsolidinterfaces ·...

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