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LECTURE NOTES ON COMPOSITE MATERIALS
SOLID MECHANICS AND ITS APPLICATIONS
Series Editor: G.M.L. GLADWELLDepartment of Civil EngineeringUniversity of WaterlooWaterloo, Ontario, Canada N2L 3GI
Aims and Scope of the SeriesThe fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchersgiving vision and insight in answering these questions on the subject of mechanics as itrelates to solids.
The scope of the series covers the entire spectrum of solid mechanics. Thus it includesthe foundation of mechanics; variational formulations; computational mechanics;statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids andstructures; dynamical systems and chaos; the theories of elasticity, plasticity andviscoelasticity; composite materials; rods, beams, shells and membranes; structuralcontrol and stability; soils, rocks and geomechanics; fracture; tribology; experimentalmechanics; biomechanics and machine design.
The median level of presentation is the first year graduate student. Some texts aremonographs defining the current state of the field; others are accessible to final year
Volume 154
undergraduates; but essentially the emphasis is on readability and clarity.
For other titles published in this series, go to www.springer.com/series/6557
Lecture Notes on Composite Materials
TOMASZ SADOWSKI
Lublin University of Technology Lublin, Poland
Edited by
Eindhoven University of Technology
Current Topics and Achievements
and
Eindhoven, The Netherlands
BORST RENÉ de
123
c© 2008 Springer Science + Business Media B.V.No part of this work may be reproduced, stored in a retrieval system, or transmittedin any form or by any means, electronic, mechanical, photocopying, microfilming, recordingor otherwise, without written permission from the Publisher, with the exceptionof any material supplied specifically for the purpose of being enteredand executed on a computer system, for exclusive use by the purchaser of the work.
Printed on acid-free paper.
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Editors
Cover illustration: WMXDesign GmbH
Tomasz SadowskiDepartment of Applied MechanicsFaculty of Mechanical EngineeringTechnical University Lublinul. Nadbystrzycka 3620-618 LublinPoland
archimedes.pol.lublin.pl
Borst de RenéDepartment of Mechanical EngineeringEindhoven University of TechnologyEindhovenThe Netherlands
ISBN 978-1-4020-8771-4 e-ISBN 978-1-4020-8772-1
Library of Congress Control Number: 2008932228
Preface
Composite materials are heterogeneous by nature, and are intended to be,since only the combination of different constituent materials can give themthe desired combination of low weight, stiffness and strength. At present, theknowledge has advanced to a level that materials can be tailored to exhibitcertain, required properties. At the same time, the fact that these materialsare composed of various, sometimes very different constituents, make theirmechanical behaviour complex. This observation holds with respect to thedeformation behaviour, but especially with respect to the failure behaviour,where complicated and unconventional failure modes have been observed.
It is a challenge to develop predictive methods that can capture this complexmechanical behaviour, either using analytical tools, or using numerical meth-ods, the finite element method being the most widespread among the latter.In this respect, developments have gone fast over the past decade. Indeed, wehave seen a paradigm shift in computational approaches to (composite) mate-rial behaviour. Where only a decade ago it was still customary to carry outanalyses of deformation and failure at a macroscopic level of observation only– one may call this a phenomenological approach – nowadays this approach isbeing progressively replaced by multiscale methods. In such methods it is rec-ognized a priori that the overall behaviour is highly dependent on local detailsand flaws. For instance, local imperfections in spacing and direction of fibrescan be detrimental to the overall bearing capacity of a structure that is com-posed of such a fibre-reinforced composite material. By upscaling, homoge-nization or methods that in a single calculation take into account the behaviourat different scales, an attempt is made to design numerical methods that havea wider range of applicability – by less reliance on adhoc assumptions – andare better rooted in the true physical behaviour of the constituent materials.Yet, few monographs have been published that present an account of recentdevelopments in the analytical/numerical modelling of composite materials.
This volume – which has grown out of a series of lectures that has been givenat Lublin University of Technology within the framework of the European
vi Preface
Community Marie-Curie Transfer-of-Knowledge project Modern CompositeMaterials Applied in Aerospace, Civil and Sanitary Engineering: TheoreticalModelling and Experimental Verification (contract MTKD-CT-2004-014058)– aims to fill this gap. It starts by a comprehensive account of methods thatcan be used at macroscopic level, followed by a précis of recent developmentsin modelling the failure behaviour of composites at a mesoscopic scale. Goingdown further, the third chapter treats fundamental concepts in micromechanicsof composite materials, including the essential concept of the RepresentativeVolume Element and Eshelby’s method. As recognized widely, failure is sel-dom a consequence of pure mechanical loadings. Often, thermal effects andlong-term effects for instance due to hygric or chemical actions play an impor-tant role as well. For this reason the ensuing two chapters are devoted to ther-mal shocks and the numerical treatment of diffusion phenomena in addition tomechanical loadings when describing failure in heterogeneous materials. Thevolume is completed by a review of fracture mechanics tools for use in theanalysis of failure in composite materials.
Eindhoven and Lublin,René de Borst and Tomasz Sadowski
Contributing Authors
Holm Altenbach is a Full Professor of Engineering Mechanics and theDirector of the Center of Engineering Sciences at the Martin-Luther-UniversitätHalle-Wittenberg (Germany). His research interests are focussed on the fol-lowing topics: Structural Mechanics (beams, plates and shells), LightweightStructures (laminates and sandwiches), Continuum Mechanics (basics and con-stitutive modelling) and Creep-damage Analysis. Since 2005 he has becomeone of the Editors-in-Chief of the Zeitschrift für angewandte Mathematik undMechanik.
René de Borst is a Distinguished Professor at Eindhoven University ofTechnology and a member of the Royal Netherlands Academy of Arts and Sci-ences. His current research interests are in the development of novel numericalmethods for the analysis of multiscale phenomena, multiphysics problems, andevolving discontinuities.
Eduard Marius Craciun is a Professor at the Faculty of Mathematicsand Informatics at the “Ovidius” University of Constanta (Romania). Mainresearch field represents the analytical and numerical methods in the study ofcrack propagation (incremental values of stresses producing crack propagationand crack propagation direction) in prestressed elastic composites and in pre-stressed and prepolarized piezoelectric materials.
Ryszard Pyrz is Professor and Chair in Materials Science and Engineer-ing and a head of the Center for New Era of Materials Technology (NEMT)at the University of Aalborg (Denmark). His scientific interest and researchactivities comprise following areas: molecular modelling of nanomaterials andnanostructures, development of a direct connection between processing andmicrostructure of advanced materials, multiscale modelling methodologies,experimental in situ investigation of materials’ microstructure with utilizationof modern techniques such as Raman nano/microspectroscopy and scanningprobe microscopy.
viii Contributing Authors
Tomasz Sadowski is the Head of the Department of Solid Mechanics at theLublin University of Technology (Poland). Prof. Sadowski’s research interestscomprise the following areas: continuum damage mechanics of materials andstructures, modelling of ceramic polycrystalline materials, modelling of com-posites with ceramic and polymer matrix, fracture mechanics of materials undermechanical loading and thermal shock.
Contents
Analysis of homogeneous and non-homogeneous plates 1H. Altenbach
1 Classification of structural models 11.1 Introductional remarks 21.2 Two-dimensional structures – definition, applications, some
basic references 31.3 Formulation principles, historical remarks 5
2 Classical plate theories 72.1 Small deflections 72.2 Large deflections 82.3 Kirchhoff plate 92.4 Mindlin plate 132.5 Von Kármán plate 14
3 Laminates and sandwiches 144 A direct approach to plate theory 19
4.1 Motivation 194.2 Direct approach for plates and shells 214.3 Tensors and their symmetry groups 25
5 Final remarks 32References 33
Numerical methods for the modelling of debonding in composites 37R. de Borst
1 Introduction 382 Levels of observation 403 Three-dimensional framework 424 Zero-thickness interface elements 455 Solid-like shell formulation 526 Meshfree methods 547 The partition-of-unity concept 618 Delamination in a solid-like shell element 689 Discontinuous Galerkin methods 70References 74
x Contents
Micromechanics of composites 77R. Pyrz
1 Introduction 772 Representative Volume Element 793 The Eshelby Equivalent Inclusion Method 82
3.1 Average Strain and Stress Theorems 843.2 Relation Between Averages 863.3 The Eshelby Solution 873.4 Equivalent Inclusion Method 91
4 The Mori–Tanaka Theory 94References 98
Non-symmetric thermal shock in ceramic matrix composite (CMC) materials 99T. Sadowski
1 Introduction to ceramic and metal matrix composites 992 Thermomechanical properties of CMCs and MMCs 102
2.1 Two-phase CMC with different elastic components 1042.2 Two-phase CMCs with plastic inclusions and MMCs
with elastic inclusions 1052.2.1 Thermomechanical properties 1052.2.2 Modelling of the whole stress–strain curve
for MMC 1062.2.3 Constitutive equations for FGMs in 3-D
formulation by self-consistent approach 1083 Temperature field under transient thermal loading 110
3.1 FEA approach for heat transfer equation (26) 1123.2 FD approach for heat transfer equation (26) 113
4 Transient thermal stress state 1155 Thermal residual stress due to technological cooling process 116
5.1 Analytical models 1165.2 Numerical models 120
6 Basic fracture mechanics concepts in functionally gradedmaterials (FGM) 1216.1 Rule of mixture to estimate the fracture toughness in FGMs 1256.2 Crack-bridging approach to assessing fracture toughness 126
7 Non-symmetric thermal shock in monolithic ceramic and FGM strip 1277.1 Transient temperature distribution during thermal shock 1297.2 Thermal and residual stresses 1297.3 Thermal stress intensity factor 1307.4 Numerical example 130
8 Two-dimensional thermal shock problem in layered circular plates 1358.1 Samples preparation and experimental procedure 1358.2 Theoretical formulation 136
8.2.1 Thermal and mechanical properties of monolithicand FGM 136
8.2.2 Heat conduction problem in FGM circular platespecimens 137
8.2.3 Determination of the thermal stress 1398.3 Numerical example 141
Contents xi
9 Concluding remarks 144Acknowledgement 144References 145
A Precis of two-scale approaches for fracture in porous media 149R. de Borst, J. Rethore, M.-A. Abellan
1 Introduction 1502 Balance equations 1513 Constitutive equations 1534 Weak form of the balance equations 1545 Micro–macro coupling 1566 Discontinuities in a two-phase medium 1597 Examples: stationary and propagating cracks 1638 Concluding remarks 168References 170
Initial deformations on behaviour of elastic composites 173E.M. Craciun
1 Homogeneous initial deformations 1732 Representation of the incremental fields 1783 The opening, sliding and tearing modes 1794 Asymptotic behavior of the incremental fields 181
4.1 The first mode 1824.2 The second mode 1834.3 The third mode 184
5 Griffith’s criterion and crack propagation 1855.1 The first opening mode 186
Acknowledgements 191References 192
Energy criteria for crack propagation in pre-stressed elastic composites 193E.M. Craciun
1 Interaction of two unequal cracks in a pre-stressed fiberreinforced composite 1931.1 Unequal cracks in a pre-stressed fiber reinforced composite.
The first fracture mode 1941.2 Asymptotic expressions. Griffith-Irwin’s method 2041.3 Crack interaction 2101.4 Pre-stressed elastic composite with two unequal cracks.
Second fracture mode 2161.5 Remarks 220
2 Sih’s generalized fracture criterion for pre-stressed orthotropic andisotropic materials 2212.1 Sih’s generalized fracture criterion for pre-stressed
orthotropic materials 2222.2 Crack propagation for orthotropic materials 225
xii Contents
2.3 Sih’s material parameterSc versus Griffith’s specific surfaceenergy γ 228
2.4 Pre-stressed isotropic material 2302.5 Sih’s energy criterion for the second fracture mode 232
Acknowledgements 235References 236
