XFEM Fracture Analysis of Composites E-Book

Soheil Mohammadi

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 474

Veröffentlichungsjahr: 2012

Beschreibung

This book describes the basics and developments of the new XFEM approach to fracture analysis of composite structures and materials. It provides state of the art techniques and algorithms for fracture analysis of structures including numeric examples at the end of each chapter as well as an accompanying website which will include MATLAB resources, executables, data files, and simulation procedures of XFEM.

• The first reference text for the extended finite element method (XFEM) for fracture analysis of structures and materials
• Includes theory and applications, with worked numerical problems and solutions, and MATLAB examples on an accompanying website with further XFEM resources
• Provides a comprehensive overview of this new area of research, including a review of Fracture Mechanics, basic through to advanced XFEM theory, as well as current problems and applications
• Includes a chapter on the future developments in the field, new research areas and possible future applications of the method

Leseprobe

Contents

Cover

Title Page

Copyright

Dedication

Preface

Nomenclature

Chapter 1: Introduction

1.1 Composite Structures

1.2 Failures of Composites

1.3 Crack Analysis

1.4 Analytical Solutions for Composites

1.5 Numerical Techniques

1.6 Scope of the Book

Chapter 2: Fracture Mechanics, A Review

2.1 Introduction

2.2 Basics of Elasticity

2.3 Basics of LEFM

2.4 Stress Intensity Factor, K

2.5 Classical Solution Procedures for K and G

2.6 Quarter Point Singular Elements

2.7 J Integral

2.8 Elastoplastic Fracture Mechanics (EPFM)

Chapter 3: Extended Finite Element Method

3.1 Introduction

3.2 Historic Development of XFEM

3.3 Enriched Approximations

3.4 XFEM Formulation

3.5 XFEM Strong Discontinuity Enrichments

3.6 XFEM Weak Discontinuity Enrichments

3.7 XFEM Crack-Tip Enrichments

3.8 Transition from Standard to Enriched Approximation

3.9 Tracking Moving Boundaries

3.10 Numerical Simulations

Chapter 4: Static Fracture Analysis of Composites

4.1 Introduction

4.2 Anisotropic Elasticity

4.3 Analytical Solutions for Near Crack Tip

4.4 Orthotropic Mixed Mode Fracture

4.5 Anisotropic XFEM

4.6 Numerical Simulations

Chapter 5: Dynamic Fracture Analysis of Composites

5.1 Introduction

5.2 Analytical Solutions for Near Crack Tips in Dynamic States

5.3 Dynamic Stress Intensity Factors

5.4 Dynamic XFEM

5.5 Numerical Simulations

Chapter 6: Fracture Analysis of Functionally Graded Materials (FGMs)

6.1 Introduction

6.2 Analytical Solution for Near a Crack Tip

6.3 Stress Intensity Factor

6.4 Crack Propagation in FGM Composites

6.5 Inhomogeneous XFEM

6.6 Numerical Examples

Chapter 7: Delamination/Interlaminar Crack Analysis

7.1 Introduction

7.2 Fracture Mechanics for Bimaterial Interface Cracks

7.3 Stress Intensity Factors for Interlaminar Cracks

7.4 Delamination Propagation

7.5 Bimaterial XFEM

7.6 Numerical Examples

Chapter 8: New Orthotropic Frontiers

8.1 Introduction

8.2 Orthotropic XIGA

8.3 Orthotropic Dislocation Dynamics

8.4 Other Anisotropic Applications

References

Index

This edition first published 2012 © 2012 John Wiley & Sons, Ltd

Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom

For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com.

The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.

Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought.

MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book's use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.

Library of Congress Cataloging-in-Publication Data

Mohammadi, S. (Soheil) XFEM fracture analysis of composites / Soheil Mohammadi. pages cm Includes bibliographical references and index. ISBN 978-1-119-97406-2 1. Composite materials–Fracture. 2. Composite materials–Fatigue. 3. Fracture mechanics. 4. Finite element method. I. Title. TA418.9.C6M64 2012 620.1′186–dc23 2012016776

A catalogue record for this book is available from the British Library.

ISBN: 978-1-119-97406-2

To: Mansoureh, Sogol & Soroush

Preface

A decade after its introduction, the extended finite element method (XFEM) has now become the primary numerical approach for analysis of a wide range of discontinuity applications, including crack propagation problems. The simplicity of the idea of enrichment for reproducing a singular/discontinuous nature of the field variable, the flexibility in handling several cracks and crack propagation patterns on a fixed mesh, and the level of accuracy with minimum additional degrees of freedom (DOFs) have transformed XFEM into the most efficient computational approach for handling various complex discontinuous problems. Concepts of XFEM are now even taught in a number of postgraduate courses, for instance advanced fracture mechanics and meshless methods, in major engineering departments, such as Civil, Mechanical, Material, Aerospace and so on, all over the world.

On the other hand, the highly flexible design of composites allows attractive prescribed tailoring of material properties, fitted to the engineering requirements for a wide range of engineering and industrial applications; from advanced aerospace and defence systems to traditional structural strengthening techniques, and from large scale turbines and power plants to nanoscale carbon nanotubes (CNTs) applications. Despite excellent characteristics, composites suffer from a number of shortcomings, mainly in the form of unstable cracking which can be initiated and propagated under different production imperfections and service circumstances. Therefore, the study of the crack stability and load bearing capacity of these types of structures, which directly affect the safety and economics of many important industries, has become one of the important topics of research for the computational mechanics community.

This text is dedicated to discussing various aspects of the application of the extended finite element method for fracture analysis of composites on the macroscopic scale. Nevertheless, most of the discussed subjects can be similarly used for fracture analysis of other materials, even on microscopic scales. The book is designed as a textbook, which provides all the necessary theoretical bases before discussing the numerical issues.

The book can be classified into four parts. The first part is dedicated to the basics. The introduction chapter provides a general overview of the problem in hand and summarily reviews available analytical and numerical techniques for fracture analysis of composites. The second chapter deals with the basics of the theory of elasticity, and is followed by discussions on asymptotic solutions for displacement and stress fields in different fracture modes, basic concepts of stress intensity factors, energy release rate, various forms of the J contour integral and mixed mode fracture criteria.

The second part, Chapter , is a redesigned and completed edition of the same chapter in my previous book, and presents a detailed discussion on the extended finite element method. After presenting the basic formulation, the chapter continues with three sections on available options for strong discontinuity enrichment functions, weak discontinuity enrichments for material interfaces, and a collection of several crack-tip enrichments for various engineering applications. It concludes with sample simulations of a wide range of problems, including classical in-plane mixed mode fracture mechanics, cracking in plates and shells, simulation of shear band creation and propagation, self-similar fault rupture, sliding contact, hydraulic fracture, and dislocation dynamics, to assess the accuracy, performance, robustness and efficiency of XFEM.

The main part of the book includes four comprehensive chapters dealing with various aspects of fracture in composite structures. Static crack analysis in orthotropic materials, dynamic fracture mechanics for stationary and moving cracks, inhomogneous functionally graded materials and bimaterial delamination analysis are discussed in detail. After a review of anisotropic and orthotropic elasticity, all chapters begin with a complete discussion of available analytical solutions for near crack-tip fields in the corresponding orthotropic problem, followed by orthotropic mixed mode fracture mechanics and associated forms of the interaction integral. XFEM enrichment functions for each class of orthotropic materials are obtained and numerical issues related to XFEM discretization are addressed. A number of illustrative numerical simulations are presented and discussed at the end of each chapter to assess the performance of XFEM compared to alternative analytical and numerical techniques.

The final part reviews a number of ongoing research topics for orthotropic materials. First, the orthotropic version of the extended isogeometric analysis (XIGA) is presented by briefly explaining the basic concepts of NURBS and IGA methodology and discussing a number of simple isotropic and orthotropic simulations. Then, the newly developed idea of plane strain anisotropic dislocation dynamics is briefly presented and related XFEM formulation and necessary enrichment functions for the self-stress state of edge dislocations are explained. The book concludes with two brief introductory sections on orthotropic biomaterial applications of XFEM and the piezoelectric problems.

I would like to express my sincere gratitude to Prof. T. Belytschko, for his valuable friendly comments and encouraging message after the publication of the first book on XFEM, and to Prof. A.R. Khoei, as a friend, a colleague and a referee with excellent comments and discussions on various subjects of computational mechanics, especially XFEM. In preparing the present book, particularly the first two parts, I have used the available contributions from brilliant research works by many others, and I have done my best to properly and explicitly acknowledge their achievements within the text, relevant figures, tables and formulae. I am much indebted to their outstanding works, and I apologize sincerely for any unintentional failure in appropriately acknowledging them.

My special thanks to many of my former and present M.Sc. and Ph.D. students who have endeavored many aspects of XFEM over the past decade. First, Dr A. Asadpoure, with whom we started to explore XFEM and new orthotropic enrichment functions in 2002. The results for the dynamic fracture of stationary and moving cracks were obtained by Dr D. Motamedi, and Ms S. Esna Ashari developed the orthotropic bimaterial enrichment functions and simulated all the delamination and interlaminar crack problems. Most of the results for FGM problems were prepared by Mr H. Bayesteh, who actively contributed in many other parts of the book. Mr S.S. Ghorashi and Mr N. Valizadeh skillfully developed the XIGA methodology and Mr S. Malekafzali implemented XFEM for anisotropic dislocation dynamics. Other results were obtained by my students: S.H. Ebrahimi, A. Daneshyar, M. Parchei, M. Goodarzi, S.N. Rezaei, S.N. Mahmoudi and M.M.R. Kabiri.

My acknowledgement is extended to John Wiley & Sons, Ltd., Publication for the excellent professional work that facilitated the whole process of publication of the book; in particular to Dr E.F. Kirkwood and E. Willner, D. Cox, R. Davies, L. Wingett, A. Hunt, C. Lim, S. Sharma and Dr R. Whitelock.

The inspiration and power for completing this work have been the love and understanding of my family, as they had to comply with all my commitments. After a life-time engagement in mathematics, physics and engineering, satisfaction is not obtained just in academic or professional progress, novelty and innovation; it should perhaps be sought in ethics, responsibility, love and freedom. This book has been completed at the twilight of a long hard winter, with a hope for a bright flourishing spring of prosperity and freedom to come. I would like to proudly dedicate this work to all spirited noble Iranian students who accomplish academic achievements while challenging for more DOFs!

Soheil MohammadiSpring 2012Tehran, IRAN

Nomenclature

Parameters not shown in this nomenclature are temporary variables or known constants, defined immediately when cited in the text.

αCurvilinear coordinateαFirst Dundurs parameterα, βNewmark parametersα, β, γFGM constantsCurvilinear coordinate α of an ellipseComponents of coordinate transformation tensorβCurvilinear coordinateβSecond Dundurs parameterCurvilinear coordinate β of an ellipse, Dilatational and shear wave functionsγWedge angleSurface energy density, Dilatational and shear wave functionsEngineering shear strainδPlastic crack tip zoneδVariation of a functionDirac delta functionKronecker delta function, Local displacements of crack edgesStrain tensorεOscillation index, Strain componentsDimensionless angular geometric functionAuxiliary strain componentsApplied displacement loadingYield strainξLocal curvilinear (mapping) coordinate systemKnot iCrack-tip positionDistance functionGauss point position along the contour JηLocal curvilinear (mapping) coordinate systemηEquivalent inelastic strainθCrack propagation angle with respect to initial crackθAngular polar coordinateCrack angle, Orthotropic angular functions, Dynamic distance functionsκ, Material parametersEffective material parameterλLame modulusλPower of radial enrichmentλRatio of orthotropic Young modules E2/E1λ, Roots of the characteristic equationμ, Isotropic and orthotropic shear modulusν, Isotropic and orthotropic Poisson's ratiosAverage orthotropic Poisson's ratiosρRadius of curvatureρDensityStress tensorApplied normal tractionCritical stress for crackingvon Mises effective stress, Stress componentsDimensionless angular geometric functionAuxiliary stress componentsYield stressHoop stressApplied tangential tractionDecohesive shear stressLevel set functionComplex stress functionAngle of orthotropic axesCrack angleRamp function for transition domainElectric potentialEnrichment function for weak discontinuitiesComplex stress functionψFriction coefficientψPhase angleEnrichment functionLevel set functionComplex stress functionBoundaryInfinitesimally small internal contourCrack boundaryTraction (natural) boundaryDisplacement (essential) boundaryFinite variation of a functionTime-stepCrack length incrementTime interval shape functionsKnot vectorΠPotential energyΦAiry stress functionMLS shape functionsComplex functionsΩDomain, Non-overlapping subdomainsDomain associated with the partition of unityDislocation glide enrichment(1, 2)Material axesaCrack length/half lengthaSemi-major axis of ellipseEffective crack lengtha(x)Vector of unknown coefficientsa, ahHeaviside enrichment degrees of freedomai, akEnrichment degrees of freedomaenrEnrichment degrees of freedomaArea associated with the domain J integralA1Area inside the infinitesimally small internal contour A+, A−Area of the influence domain above and below the crackAi, AijCoefficientsbWidth of a platebSemi-minor axis of ellipsebk, blkCrack tip enrichment degrees of freedomBurgers vector for dislocation αMagnitude of the Burgers vector for dislocation αbnSeries coefficientsBMatrix of derivatives of shape functionsB12, B66Coefficients of characteristic equationB-spline basis function of order pBhMatrix of derivatives of final shape functionsBriStrain-displacement matrix (derivatives of shape functions)BuiMatrix of derivatives of classical FE shape functionsBaiMatrix of derivatives of Heaviside enrichment shape functionsBbiMatrix of derivatives of crack tip enrichment shape functionsBciMatrix of derivatives of weak discontinuity enrichment shape functionsBciMatrix of derivatives of transition shape functionscDugdale effective crack lengthcJSize of crack tip contour for J integralcijMaterial compliance constantscmDegrees of freedom for weak discontinuity enrichmentcmDegrees of freedom for transitional enrichmentcdDilatational wave speedcLWave speed along the loading axiscRRayleigh speedcsShear wave speedcMaterial constitutive matrix4th order material compliance tensorCijklCartesian components of CnCoefficientNURBS curvedijMaterial modulus constantsddijDynamic material modulus constantsdmDegrees of freedom for transitional enrichmentDDynamic functionDTwo dimensional Material modulus matrix4th order material elasticity modulus tensorCartesian components of Components of DDi, Dx, DyElastic displacement vectorE, EiIsotropic and orthotropic Young's modulesEi, Ex, EyElectric fieldEffective material parameterE0Reference Young modulusE12Equivalent bimaterial elastic modulusAverage orthotropic Young modulesfk(x)Set of PU functionsfNodal force vectorfriNodal force components (classic and enriched)fbBody force vectorftExternal traction vectorfcCohesive crack traction vectorfextExternal force vectorfI, fIIFunctions of the crack-tip speedfdI, fdIIUniversal functions of the crack-tip speedfIij, fIIij, fIIIijMode I, II and II angular functionsfpukSet of PU functionsFl(x), Crack tip enrichment functionsDelamination functionDeformation gradientAngular function for a crack-tip kink problem, Orthotropic crack-tip enrichment functionsGShear modulusG, Fracture energy release rateGcCritical fracture energy release rateG1, G2Mode I and II fracture energy release rateshIntrinsic shear band thicknesshtCharacteristic thickness of the bonding layerhSlope of linear softening curveIntrinsic hardening coefficient, H(x)Heaviside functionComplex number I2nd order identity tensor4th order symmetric identity tensorI(t)Corresponding creep complianceJJacobian matrixJ, JsJ integralJactActual J integralJauxAuxiliary J integralJ1, J2Components of the J vectorJdkDynamic J integralk0Dimensionless constant for the power hardening lawKBulk modulusKStiffness matrixKrsijComponents of stiffness matrixKStress intensity factorK, Complex stress intensity factorKcCritical stress intensity factorK0Reference stress intensity factorKI, KII, KIIIMode I, II and III stress intensity factors, Normalized mode I and mode II stress intensity factorsKauxI, KauxIIAuxiliary mode I and mode II stress intensity factorsKIc, KIIcCritical mode I and mode II stress intensity factorsK1Ic, K2IcFracture toughnesses along the principal planes of elastic symmetryFracture toughness at propagationKdIcDynamic crack initiation toughnessKdIcDynamic crack growth (propagation) toughness, KttHoop stress intensity factorlijCoefficientLLength of the singular elementL(vc)Dynamic matrix for orthotropic materialsmNumber of enrichment functionsmtNumber of nodes to be enriched by crack-tip enrichment functionsmhNumber of nodes to be enriched by Heaviside enrichment functionsmfNumber of crack-tip enrichment functionsmmNumber of weak discontinuity enrichment functionsmstNumber of transition enrichment functions 1m shNumber of transition enrichment functions 2mkRoots of characteristic equation mk=mkx+imkymConcentrated bending momentmInteraction integralM(1), M(2)Interaction integral associated with two modes I and IIMdDynamic interaction integralMMass matrixMijComponents of mass matrixnPower number for the HKK plastic modelnNumber of nodes for each finite elementngANumber of gauss points inside contour area aNumber of gauss points on contour npNumber of independent domains of partition of unitynn, nnodesNumber of nodes in a finite elementne, nelem.Number of finite elementsncpNumber of control pointsncellsNumber of background cells of EFGnDOFsNumber of degrees of freedomnNormal vectorNjMatrix of shape functionsNjShape functionNelementsNumber of finite elementsnnodesNumber of nodesNenrich.Number of enrichment functionsnDOFsNumber of degrees of freedomHierarchical shape functions for the transition domainNew set of GFEM shape functionsp(x)Basis functionpA point on curvilinear coordinate system p, pk, Orthotropic parameterspBasis functionpenrEnrichment basis functionplinLinear basis functionpkk-th basis functionpl(x)l-order polynomial functionpConcentrated forcepExternal load vectorPcrCritical loadqArbitrary smoothing functionq, qk, Orthotropic parametersqiNodal values of the arbitrary smoothing functionLocal crack tip polar coordinatesrJRadius of J integral contourrd, rsDilatational and shear distance functionsrl, rsOrthotropic distance functionsrp, rp1, rp2Crack tip plastic zoneRRamp functionRKRatio of dynamic stress intensity factorsRatio of opening to sliding displacementsNURBS function of order psRoots of characteristic equation s=s1+is2skRoots of characteristic equation sk=skx+iskyRoots of characteristic equation smRoots of characteristic equation sm=sm1+ism2Slope of softening curveSijMaterial constantsNURBS surfacetTimetTractionthUnit vector for tangential directiontijMaterial functiont0Time for the wave to reach the crack tiptpFRP thicknessTi(t)Time shape functionsTjEnriched time intervalTjTransformation matrixTiControl pointsuDisplacement vectorVelocity vectorPrescribed displacementPrescribed velocityAcceleration vectoruiDisplacement field componentVelocity field componentAcceleration field componentuauxiAuxiliary displacement field componentAuxiliary velocity field componentuenrEnriched displacement fielduFEClassical finite element displacement fielduXFEMXFEM displacement fieldutraTransition enrichment part of the displacement fielduHHeaviside enrichment part of the displacement fieldutipCrack-tip enrichment part of the displacement fieldumatWeak discontinuity enrichment part of the displacement fielduEnrtip(x, t)Crack-tip part of the approximationuhnDisplacement at time nVelocity at time nAcceleration at time nuh, uh(x)Approximated displacement fieldApproximated velocity fieldApproximated acceleration fieldNodal displacement vectorDisplacement angular functionsux, uy, uzx, y and z displacement components, Local displacements of the nodes along the crack in the singular elementUI, UIISymmetric and antisymmetric crack tip displacementsUsStrain energyUes, UpsElastic and plastic strain energiesSurface energyvcCrack-tip velocityvVelocity vectorvcClassical velocity DOFsveAdditional velocity DOFsV(t)Vector of approximated velocity degrees of freedomVVolumeWExternal workwauxAuxiliary workWextVirtual work of the external loadingWgGauss weight factorWiWeights associated with each control point iGauss weighting factor for contour WAgGauss weighting factor for area inside contour integral JWintInternal virtual workwMInteraction workwdKinetic energy densitywsStrain energy densityWt(t)Time weight functionx, y, zCartesian coordinates(X, Y)Global coordinate system(X1, X2)Global coordinate systemxPosition vectorxcPosition of crack or discontinuityxtipPosition of the crack tipPosition of the projection point on an interface(X1, X2)2D coordinate system(X1, X2)Material axesLocal crack tip coordinate axeszComplex variable z=x+iyConjugate complex variable ziComplex parametersTime derivativeMaterial time derivative, The first and second temporal derivatives of a function, The first and second spatial derivatives of a function, The first and second integrals of a functionNabla operatorJump operator across an interface:Inner product of two second order tensorsTensor product of two vectorsReal part of a complex numberImaginary part of a complex numberProportionalBEMBoundary element methodCADComputer aided designCNTComposite nanotubeCODCrack opening displacementCTODCrack-tip opening displacementDCTDisplacement correlation techniqueDEMDiscrete element methodDOFDegree of freedomEDIEquivalent domain integralEFGElement free GalerkinELMEquilibrium on lineEPFMElastic plastic fracture mechanicsFDMFinite difference methodFEFinite elementFEMFinite element methodFGMFunctionally graded materialsFMMFast marching methodFPMFinite point methodFRPFibre reinforced polymerGFEMGeneralized finite element methodGNpjGeneralized Newmark approximation of degree p for equations of order jHRRHutchinson-Rice-RosengrenIGAIsogeometric analysisLEFMLinear elastic fracture mechanicsLSMLevel set methodMCCModified crack closureMLPGMeshless local Petrov GalerkinMLSMoving least squaresNURBSNon-uniform rational B-splinesOUMOrdered upwind methodPUPartition of unityPUFEMPartition of unity finite element MethodRKPMReproducing kernel particle methodSARStatically admissible stress recoverySIFStress intensity factorSPHSmoothed particle hydrodynamicsSSpjSingle step approximation of degree p for equations of order jSTXFEMSpace time extended finite element methodTXFEMTime extended finite element methodWLSWeighted least squaresXFEM, X-FEMExtended finite element methodXIGAExtended isogeometric analysis

1

Introduction

1.1 Composite Structures

Composite materials are used extensively in engineering and industrial applications; from traditional structural strengthening to advanced aerospace and defence systems, and from nanoscale carbon nanotube (CNT) applications to large scale turbines and power plants and so on. Their highly flexible design allows prescribed tailoring of material properties, fitted to the engineering requirements. These include a wide variety of properties across various length scales, including nano and micro-mechanical structural needs, thermo-mechanical specifications and even electro-magneto-mechanical characteristics. In addition, multilayer and orthotropic functionally graded materials (FGMs) have been increasingly used in advanced material systems in high-tech industries to withstand hostile operating conditions, where conventional homogeneous composites may fail.

Composite materials are created by the combination of two or more materials to form a new material with enhanced properties compared to those of the individual constituents. By this definition, reinforced concrete, as a mixture of stone, sand, cement and steel, wood comprised of cellulose and lignin, and bone consisting of collagen and apatite can be regarded as special types of composites. The conventional forms of composites, however, are made of two main ingredients: fibres and matrix. Fibres are required to have a number of specifications, such as high elasticity modulus and ultimate strength, and must retain their geometrical and mechanical properties during fabrication and handling. The matrix constituent must be chemically and thermally compatible with the fibres over a long period of time, and is meant to bind together the fibres, protect their surfaces, and transfer stresses to the fibres efficiently.

High specific strength, excellent fatigue durability, significant corrosion, chemical and environmental resistances, especially important in food and chemical processing plants, cooling towers, offshore platforms and so on, designable mechanical properties, electromagnetic transparency or electrical insulation, together with relatively fast deployment and low maintenance have made composites attractive materials for almost all engineering applications.

Despite excellent characteristics, composites suffer from a number of shortcomings, such as brittleness, high thermal and residual stresses, poor interfacial bonding strength and low toughness, which may facilitate the process of unstable cracking under different conditions, such as imperfection in material strength, fatigue, yielding and production faults. These failures can cause extensive damage accompanied by substantial reduction in stiffness and load bearing capacity, decreased ductility and the possibility of abrupt collapse mechanisms. The problem becomes even more important in intensive concentrated loading conditions, such as moving and dynamic loadings, high velocity impact and explosion.

Moreover, composite materials are utilized in thin forms which are susceptible to various types of defects. Cracking, the most likely type of defect in these structures, can be initiated and propagated under different production imperfections and service circumstances, such as initial weakness in material strength, fatigue and yielding. Therefore, the study of the crack stability and load bearing capacity of these types of structures, which directly affect the safety and economics of many important industries, has become an important topic of research for the computational mechanics community.

1.2 Failures of Composites

Layered, orthotropic, sometimes inhomogeneous and multi-material characteristics of composites allow the possibility for occurrence of various failure modes under different loading conditions. In general, however, the failure modes of composite plies can be categorized into four classes: fibre failure, ply delamination, matrix cracking and fibre/matrix deboning. These failure modes, or any combination of them, reduce and may ultimately eliminate the composite action altogether.

1.2.1 Matrix Cracking

The matrix material is the lowest strength component in a composite action to withstand a specific loading. The brittle nature of matrix cracking is the main source of failure in composites and may initiate other modes of failure, such as delamination and debonding.

1.2.2 Delamination

Delamination, also called interlaminar debonding or interface cracking, is among the most commonly encountered failure modes in composite laminates and may become a major source of concern in the performance and safety of composites by reducing the ductility, stiffness and strength of the composite specimen and even cause sudden brittle fracture mechanisms.

Delaminations can be initiated or extended from high stress concentrations that originate from mechanical effects, such as manufacturing, transportation and service effects, such as temperature, moisture, matrix shrinkage, or from general loading conditions, especially sudden concentrated loadings such as impact and explosion.

These effects may become more severe around curved sections, sudden changes of cross sections, and free edges. One important aspect of delamination failure is that substantial internal damage may exist in the interface adjacent plies without any apparent external destruction.

1.2.3 Fibre/Matrix Debonding

A perfect bonding between the fibre and matrix is necessary to ensure the composite action. Any debonding, or even local sliding, may substantially affect the overall strength of the composite specimen. It is generally accepted that the composite material should be designed and manufactured in such a way that fibre/matrix debonding never occurs before matrix cracking and delamination.

1.2.4 Fibre Breakage

Fibre breakage is probably the last mode of failure of a composite specimen prior to its collapse. Once the fibres are broken, the load bearing capacity of the specimen suddenly drops to almost zero.

1.2.5 Macro Models of Cracking in Composites

Homogeneous composites are primarily assumed to behave in an orthotropic linear elastic state. Equivalent homogeneous orthotropic material properties are determined based on the assumption of an equivalent smeared fibre/matrix mixture. This is certainly the case for most numerical solutions at the macroscopic level. As a result, fracture is assumed to occur only in an in-plane cracking mode (matrix/fibre cracking) or in an interlaminar cracking state (delamination), as depicted in Figure 1.1.

Figure 1.1 Main modes of cracking in composites.

1.3 Crack Analysis

In this section, a brief review of the main available theoretical approaches for analysis of crack stability and propagation is presented. There are different classifications in crack analysis. For example, from the geometrical point of view, a crack may be represented as an internal discontinuity or external boundary (discrete crack model), characterizing a strong displacement discontinuity (Figure 1.2a), or its equivalent continuum mechanical effects (in terms of stiffness and strength reduction) can be considered within the numerical model in a distributed fashion without explicitly defining its geometry (smeared crack model) (see Figure 1.2b).

Figure 1.2 Discrete and smeared crack models in a typical finite element mesh.

1.3.1 Local and Non-Local Formulations

Early attempts to simulate crack problems by numerical methods adopted a simple rule to check the stress state at any sampling point against a material strength criterion. The constitutive behaviour of the point was only affected by its own local stress–strain state (point 1 in Figure 1.3). Soon it was realized that cracking could not be regarded solely as a local point-wise stress-based criterion, and such a local approach for fracture analysis may become size or mesh dependent and unreliable.

Figure 1.3 Local and non-local evaluation of the cracking state.

The remedy was the introduction of non-local formulations based on characteristic length scales (Bazant and Planas, 1997), defined for the material constitutive law as an intrinsic material property, or for a numerical model based on the geometrical requirements. To clarify the basic idea, consider a very simplified case (point 2 in Figure 1.3), where the fracture behaviour of this point is determined from a non-local criterion expressed in terms of the state variables (including the length scale) at that point and a number of surrounding points in its support domain.

1.3.2 Theoretical Methods for Failure Analysis

Three fundamental approaches are available for discussion of the effects of defects and failures: continuum-based plasticity and damage mechanics and the crack-based approach of fracture mechanics. All three approaches can be implemented within different numerical methods. These methods, however, are applied to fundamentally different classes of failure problems. While the theory of plasticity and damage mechanics are basically designed for problems where the displacement field, and usually the strain field, remain continuous everywhere, fracture mechanics is essentially formulated to deal with strong discontinuities (cracks) where both the displacement and strain fields are discontinuous across a crack surface (Mohammadi, 2003a, 2008).

In practice, however, damage mechanics and the theory of plasticity have been modified and adapted for failure/fracture analysis of structures with strong discontinuities and fracture mechanics is sometimes used for weak discontinuity problems. It is, therefore, difficult to distinguish between the practical engineering applications exclusively associated with each class.

1.3.2.1 Plasticity

Plasticity theory is well developed to deal with plastic deformations and is based on various local-failure criteria, written in terms of local (point-wise) state variables, such as stress tensor and elastic and plastic strain components. Most of the plasticity based crack analyses are based on softening plasticity models of smeared cracking, which may become mesh or size dependent, if higher order formulations (such as the Cosserate or gradient theories) are not adopted. Plasticity models are capable of predicting the initiation of crack as well as predicting its growth, and can be readily implemented in different numerical techniques.

1.3.2.2 Fracture Mechanics

In contrast, the theory of linear elastic fracture mechanics (LEFM) is based on the existence of an initial crack or flaw and adopts the laws of thermodynamics to formulate an energy-based criterion for analysis of the existing crack. Such an approach ensures the size-independence of the solution. Clearly, the method is based on an explicit discrete definition of crack in the form of internal or external boundaries. Another important aspect of LEFM is its capability in deriving the singular stress field, predicted by the analytical solution at a crack tip. These two specifications have substantially complicated the numerical techniques designed for fracture mechanics analysis.

In addition to original linear elastic formulation, fracture mechanics has been extended to limited nonlinear behaviour and plasticity around the crack tip, forming the theory of elastoplastic fracture mechanics (EPFM).

1.3.2.3 Damage Mechanics

Damage mechanics has been increasingly adopted to analyze failure in various engineering application involving concrete, rock, metals, composites, and so on. Damage mechanics is a non-local approach (similar to fracture mechanics) but with a formulation apparently similar to the softening theory of plasticity. In damage mechanics, both the strength and stiffness of a material point are decreased if it experiences some level of damage. This is in contrast to the classical theory of plasticity, where the stiffness remains unchanged and only the strength is updated according to the hardening/softening behaviour.

Thermodynamics principles are adopted to derive the necessary formulation based on the micro-cracking state of the material and one of the fundamental assumptions of equivalent strain or equivalent strain energy principles to relate the equivalent undamaged model with the real damaged one. Such an equivalent undamaged model holds the continuity of the model intact.

1.4 Analytical Solutions for Composites

1.4.1 Continuum Models

Conventional continuum lamination models for analysis of composites are based on a composite element, which considers the fibre/matrix mixture as an equivalent homogeneous orthotropic continuum laminate, with perfect bond between the constituents in each single lamina, no strain discontinuity across the interface, and a regular arrangement of fibres. Equilibrium equations, compatibility conditions and the linear elastic Hook's law determine the elasticity constants and govern the stress–strain constitutive law or its generalized form. The classical lamination theory formulates the multilayered laminate based on variations of fibre orientation, stacking sequence and ply-level material properties. More advanced models assume a viscoelasticity model for the matrix. A number of analytical models have also been developed to account for a number of failure modes in each ply, but they are, unfortunately, limited to very simplified geometries, and specific orientations, stacking and loading conditions.

1.4.2 Fracture Mechanics of Composites

Linear elastic fracture mechanics (LEFM) is based on the existence of a crack or a flaw and determines the state of its stability and possible propagation. Its non-local nature guarantees the size or mesh (in case of a finite element analysis) independency of the solution. Definitions of non-local concepts such as the stress intensity factor, energy release rate and energy-based criteria allow the classical fracture mechanics to be extended to nonlinear problems. Most of the research in this field, however, can be classified into four categories: static cracking in a single orthotropic material, dynamic orthotropic cracking, orthotropic bimaterial interface cracks and fracture in orthotropic FGMs. All categories include topics on definition and evaluation of stress intensity factors, associated J and interaction integrals, deriving the asymptotic solutions, crack propagation criteria, and so on.

The fracture mechanics of composite structures has been studied by many researchers. Beginning with the pioneering work by Muskelishvili (1953), several others such as Sih, Paris and Irwin (1965), Bogy (1972), Bowie and Freese (1972), Barnett and Asaro (1972), Kuo and Bogy (1974), Tupholme (1974), Atluri, Kobayashi and Nakagaki (1975a), Forschi and Barret (1976), Boone, Wawrzynek and Ingraffea (1987), Viola, Piva and Radi (1989) and, more recently, Lim, Choi and Sankar (2001), Carloni and Nobile (2002), Carloni, Piva and Viola (2003) and Nobile and Carloni (2005) have proposed solutions for various anisotropic static and quasi-static crack problems.

Simultaneously, several researchers have contributed to finding the elastodynamic fields around a propagating crack within an anisotropic medium, including Achenbach and Bazant (1975), Arcisz and Sih (1984), Piva and Viola (1988), Viola, Piva and Radi (1989), Shindo and Hiroaki (1990), De and Patra (1992), Gentilini, Piva and Viola (2004), Kasmalkar (1996) and Chen and Erdogan (1996). Lee, Hawong and Choi (1996) derived the dynamic stress and displacement components around the crack tip of a steady state propagating crack in an orthotropic material. The same subject was then followed by Gu and Asaro (1997), Rubio-Gonzales and Mason (1998), Broberg (1999), Lim, Choi and Sankar (2001), Federici et al. (2001), Nobile and Carloni (2005), Piva, Viola and Tornabene (2005) Sethi et al. (2011), and Abd-Alla et al. (2011), among others.

The research has not been limited to single layer orthotropic homogeneous composites. Analytical solutions for delamination in multilayer composites have also been investigated comprehensively. The first attempt was probably by Williams (1959) who discovered the oscillatory near-tip behaviour for a traction-free interface crack between two dissimilar isotropic elastic materials, followed by several others such as Erdogan (1963), Rice and Sih (1965), Malysev and Salganik (1965), England (1965), Comninou (1977), Comninou and Schmuser (1979), Sun and Jih (1987), Hutchinson, Mear and Rice (1987) and Rice (1988), among others. The study of interface cracks between two anisotropic materials was performed by Gotoh (1967), Clements (1971) and Willis (1971), followed by Wang and Choi (1983a, 1983b), Ting (1986), Tewary, Wagoner and Hirth (1989), Wu (1990), Gao, Abbudi and Barnett (1992) and Hwu (1993a, 1993b), Bassani and Qu (1989), Sun and Manoharan (1989), Suo (1990), Yang, Sou and Shih (1991), Hwu (1993b), Qian and Sun (1998), Lee (2000) and Hemanth et al. (2005).

Fracture mechanics of FGMs has similarly been an active topic for analytical research. For instance, Yamanouchi et al. (1990), Holt et al. (1993), Ilschner and Cherradi (1995), Nadeau and Ferrari (1999), Takahashi et al. (1993), Pipes and Pagano (1970, 1974), Pagano (1974), Kurihara, Sasaki and Kawarada (1990), Niino and Maeda (1990), Sampath et al. (1995), Kaysser and Ilschner (1995), Erdogan (1995) and Lee and Erdogan (1995) have studied various aspects of FGM properties. Despite material inhomogeneity, Sih and Chen (1980), Eischen (1983) and Delale and Erdogan (1983) have shown that the asymptotic crack-tip stress and displacement fields for certain classes of FGMs follow the general form of homogeneous materials and Ozturk and Erdogan (1997) and Konda and Erdogan (1994) analytically solved for crack-tip fields in inhomogeneous orthotropic infinite FGM problems. Evaluation of the J integral for determining the mixed-mode stress intensity factors in general FGM problems was studied by Gu and Asaro (1997), Gu, Dao and Asaro (1999), Anlas, Santare and Lambros (2000) and, in particular, Kim and Paulino (2002a, 2002b, 2000c, 2003a, 2003b, 2005) who examined and developed three independent formulations: non-equilibrium, incompatibility and constant-constitutive-tensor, for the J integral.

1.5 Numerical Techniques

Due to the limitations and inflexible nature of analytical methods in handling arbitrary complex geometries and boundary conditions and general crack propagations, several numerical techniques have been developed for solving composite fracture mechanics problems.

The finite element method has been widely used for fracture analysis of structures for many years and is probably the first choice of analysis for general engineering problems, including fracture, unless a better solution is proposed. Despite outstanding advantages, alternative methods are also available, including the adaptive finite/discrete element method (DEM), the boundary element method (BEM), a variety of meshless methods, the extended finite element method (XFEM), the extended isogeometric analysis (XIGA) and, more recently, advanced multiscale techniques. In the following, a brief review of a number of studies on fracture analysis of composites for each class of numerical methods is presented.

1.5.1 Boundary Element Method

Cruse (1988), Aliabadi and Sollero (1998) and García-Sánchez, Zhang and Sáez (2008) developed boundary element solutions for quasi-static crack propagation and dynamic analysis of cracks in orthotropic media. In the boundary element method, a number of elements are used to discretize the boundary of the problem domain, and the domain itself is analytically represented in the governing equations. The boundary element method, regardless of all the benefits, cannot be readily extended to nonlinear systems and is not suited to general crack propagation problems (Figure 1.4c).

Figure 1.4 Various numerical methods for crack analysis.

1.5.2 Finite Element Method

Most of the performed numerical analyses of structures prior to the end of the twentieth century were related to the finite element method. The finite element method can be easily adapted to complex geometries and general boundary conditions and is well developed into almost every possible engineering application, including nonlinear, inhomogeneous, anisotropic, multilayer, large deformation, fracture and dynamic problems. Several general purpose finite element softwares have been developed, verified and calibrated over the years and are now available to almost anyone who asks (and pays) for them. Furthermore, concepts of FEM are now offered by all engineering departments in the form of postgraduate and even undergraduate courses.

Introduction and fast development of the finite element method drastically changed the extent of application of LEFM from classical idealized models to complex practical engineering problems. After earlier application of FEM to fracture analysis of composites (for instance by Swenson and Ingraffea, 1988) a large number of studies adopted FEM to simply obtain the displacement, strain and stress fields required for numerical evaluation of fracture mechanics parameters such as the stress intensity factors, the energy release rate or the J integral to assess the stability of crack (Rabinovitch and Frosting, 2001; Pesic and Pilakoutas, 2003; Rabinovitch, 2004; Colombi, 2006; Lu et al., 2006; Yang, Peng and Kwan, 2006; Bruno, Carpino and Greco, 2007; Greco, Lonetti and Blasi, 2007).

Later, various techniques were developed within the FEM framework to allow reproduction of a singular stress field at a crack tip and to facilitate simulation of arbitrary crack propagations. Singular finite elements, developed to accurately represent crack-tip singular fields (Owen and Fawkes, 1983), provide the major advantage of simple construction of the model by simply moving the nearby midside nodes to the quarter points with no other changes in the finite element formulation being required (Figure 1.4a). Prior to the development of XFEM, singular elements were the most popular approach for fracture analysis of structures. Singular elements, however, have to be used in a finite element mesh, where crack faces have to match element boundaries. This largely limits their application to general crack propagation problems, unless combined with at least a local adaptive finite element scheme.

1.5.3 Adaptive Finite/Discrete Element Method

The adaptive finite element method, combined with the concepts of contact mechanics of the discrete element method (DEM), has been adopted in several studies for simulation of progressive crack propagation in composites under quasi-static and dynamic/impact loadings. They include a variety of crack models, such as smeared crack, discrete inter-element crack models, cracked interface elements and the discrete contact element, which may use general contact mechanics algorithms to simulate progressive delamination and fracture problems (Mohammadi, Owen and Peric, 1997; Sprenger, Gruttmann and Wagner, 2000; Wu, Yuan and Niu, 2002; Mohammadi and Forouzan-sepehr, 2003; Wong and Vecchio, 2003; Wu and Yin, 2003; Mohammadi, 2003b; Lu et al., 2005; Wang, 2006; Teng, Yuan and Chen, 2006; Mohammadi and Mousavi, 2007; Moosavi and Mohammadi, 2007; Mohammadi, 2008; Rabinovitch, 2008).

In a composite progressive fracture analysis, part of the composite model, which is potentially susceptible to damage, is represented by discrete elements and the rest of the specimen is modelled with coarser finite elements to reduce the analysis time (Figure 1.4f). Each discrete element can be discretized by a finite element mesh; finer for the plies closer to the damaged region and coarser elsewhere. A contact methodology controls the debonding mechanisms and all post delamination behaviour, such as sliding (Mohammadi, Owen and Peric, 1997; Mohammadi, 2008). On occurrence of a crack or after a crack propagation step based on a simple comparison of the computed stress state with the adhesive strength (Parker, 1981; O’Brien, 1985; Rowlands, 1985; Roberts, 1989; Taljsten, 1997), adaptive schemes are used to locally remesh the finite element model to ensure matching of element edges and crack faces. Nonlinear material properties and geometric nonlinearities can be considered in the basic FE formulation. The method, however, is numerically expensive and the nodal alignments may cause numerical difficulties and mesh dependency to some extent in propagation problems.

1.5.4 Meshless Methods

Meshless methods include a wide variety of numerical methods with different approximation techniques, diverse solution schemes, variable levels of accuracy and dissimilar applications. Ironically, in a general view, they have nothing in common but being different from the finite element method. The main idea of meshless methods is to avoid a predefined fixed connectivity between the nodal points which are used to define the geometry and to set necessary degrees of freedom to discretize the governing equation. As a result of such a connection-free style of nodal discretization, any existing cracks or crack propagation paths can be efficiently embedded geometrically within the numerical model (Figure 1.4d).

Meshless methods have been used extensively for analysis of various engineering problems (Belytschko et al., 2002). They include the frequently used classes of the element-free Galerkin method (EFG) (Belytschko, Lu and Gu, 1994; Belytschko, Organ and Krongauz, 1995; Ghorashi, Valizadeh and Mohammadi, 2011), the meshless local Petrov–Galerkin (MLPG) (Atluri and Shen, 2002), smoothed particle hydrodynamics (SPH) (Belytschko et al., 2000; Madani and Mohammadi, 2011; Ostad and Mohammadi, 2012), the finite point method (FPM) (Onate et al., 1995; Bitaraf and Mohammadi, 2010), isogeometric analysis (IGA) (Hughes, Cottrell and Bazilevs, 2005) and its extended version (XIGA) (Ghorashi, Valizadeh and Mohammadi, 2012), and other approaches including the reproducing kernel particle method (RKPM) (Liu et al., 1996), HP-clouds (Duarte and Oden, 1995), the equilibrium on line method (ELM) (Sadeghirad and Mohammadi, 2007) and the smoothed finite element method (Liu and Trung, 2010), among others. It is noted that some of the mentioned methods are in fact a combination of finite element and meshless concepts. By this definition, the extended finite element method (XFEM) may somehow be included.

Despite higher accuracy and flexible adaptive schemes, the majority of meshless methods are yet to be user friendly because of weak versatility to deal with arbitrary boundary conditions and geometries, complicated theoretical bases, high numerical expense and the need for sensitivity analysis, calibration and difficult stabilization schemes in many of them.

1.5.5 Extended Finite Element Method

Since the introduction of the extended finite element method (XFEM) for fracture analysis by Belytschko and Black (1999), Moës, Dolbow and Belytschko (1999) and Dolbow (1999), based on the mathematical foundation of the partition of unity finite element method (PUFEM), proposed earlier by Melenk and Babuska (1996) and Duarte and Oden (1996), XFEM methodology has been rapidly extended to a vastly wide range of applications that somehow include a local discontinuity or singularity within the solution.

The natural extension of FEM into XFEM allows new capabilities while preserving the finite element original advantages. The two main superiorities of XFEM are its capability in reproducing the singular stress state at a crack tip, and allowing several cracks or arbitrary crack propagation paths to be simulated on an independent unaltered mesh (Figure 1.4b). In fact, while the presence of the crack is not geometrically modelled and the mesh does not need to conform to the virtual crack path, the exact analytical solutions for singular stress and discontinuous displacement fields around the crack (tip) are reproduced by inclusion of a special set of enriched shape functions that are extracted from the asymptotic analytical solutions.

Apart from earlier works that were directed towards the development of the extended finite element method for linear elastic fracture mechanics (LEFM), simulation of failure and localization has been the target of several studies including, Jirásek and Zimmermann (2001a, 2001b), Sukumar et al. (2003), Dumstorff and Meschke (2003), Patzak and Jirásek (2003), Ventura, Moran and Belytschko ( 2005), Samaniego and Belytschko (2005), Areias and Belytschko (2005a, 2005b, 2006) and Song, Areias and Belytschko (2006). In addition, various contact problems have been simulated by XFEM. For example, Dolbow, Moes and Belytschko (2000c, 2001), Shamloo, Azami and Khoei (2005), Belytschko, Daniel and Ventura (2002), Khoei and Nikbakht (2006), Khoei, Shamloo and Azami (2006) and Khoei et al. (2006) have used simple Heaviside enrichments to deal with contact discontinuities in different problems, while Ebrahimi, Mohammadi and Kani (2012) have recently proposed a numerical approach within the partition of unity finite element method to determine the order of singularity at a sliding contact corner.

Moreover, several dynamic fracture problems have been studied by XFEM. Among them, Peerlings et al. (2002), Belytschko et al. (2003), Oliver et al. (2003), Ventura, Budyn and Belytschko (2003), Chessa and Belytschko (2004, 2006), Belytschko and Chen (2004), Zi et al. (2005), Rethore, Gravouil and Combescure (2005a), Rethore et al. (2005), Menouillard et al. (2006), Gregoire et al. (2007), Nistor, Pantale and Caperaa (2008), Prabel, Marie and Combescure (2008), Combescure et al. (2008), Gregoire, Maigre and Combescure (2008), Motamedi (2008), Kabiri (2009), Gravouil, Elguedj and Maigre (2009a, 2009b) and Rezaei (2010) have discussed various aspects of general dynamic fracture problems.

In the past decade, development of XFEM has substantially contributed to new studies of fracture analysis of various types of composite materials. Dolbow and Nadeau (2002) employed XFEM to simulate fracture behaviour of micro-structured materials with a focus on functionally graded materials. Then, Dolbow and Gosz (2002) described a new interaction energy integral method for the computation of mixed mode stress intensity factors in functionally graded materials. In a related contribution, Remmers, Wells and de Borst (2003) presented a new formulation for delamination in thin-layered composite structures. A study of bimaterial interface cracks was performed by Sukumar et al. (2004) by developing new bimaterial enrichment functions. Nagashima, Omoto and Tani (2003) and Nagashima and Suemasu (2004, 2006) described the application of XFEM to stress analyses of structures containing interface cracks between dissimilar materials and concluded the need for orthotropic enrichment functions to represent the asymptotic solution for a crack in an orthotropic material. Hettich and Ramm (2006) simulated the delamination crack as a jump in the displacement field without using any crack-tip enrichment. Also, a number of XFEM simulations have focused on thermo-mechanical analysis of orthotropic FGMs (Dag, Yildirim and Sarikaya, 2007), 3D isotropic FGMs (Ayhan, 2009; Zhang et al., 2011; Moghaddam, Ghajar and Alfano, 2011) and frequency analysis of cracked isotropic FGMs (Natarajan et al., 2011). Recently, Bayesteh and Mohammadi (2012) have used XFEM with orthotropic crack-tip enrichment functions to analyze several FGM crack stability and propagation problems.

Development of independent orthotropic crack-tip enrichment functions was reported in a series of papers by Asadpoure, Mohammadi and Vafai (2006, 2007), Asadpoure and Mohammadi (2007) and Mohammadi and Asadpoure (2006). Later, Motamedi (2008) and Motamedi and Mohammadi (2010a, 2010b, 2012) studied the dynamic crack stability and propagation in composites based on static and dynamic orthotropic enrichment functions and Esna Ashari (2009) and Esna Ashari and Mohammadi (2009, 2010b, 2011a, 2012) have further extended the method for orthotropic bimaterial interfaces.

1.5.6 Extended Isogeometric Analysis

Since the introduction of isogeometric analysis (IGA) by Hughes, Cottrell and Bazilevs (2005) a new and fast growing chapter has been opened in unifying computer aided design (CAD) and numerical solutions by using the non-uniform rational B-splines (NURBS) functions (Figure 1.4e). The method, in fact, can be categorized with other meshless methods, but it is briefly introduced due to its fast growing state and excellent potentials.

IGA has been successfully adopted in several engineering problems, including structural dynamics (Cottrell, Hughes and Bazilevs, 2009; Hassani, Moghaddam and Tavakkoli, 2009), Navier–Stokes flow (Nielsen et al., 2011), fluid–solid interaction (Bazilevs et al., 2009), shells (Benson et al., 2010a; Benson et al., 2011; Kiendl et al., 2010; Kiendl et al., 2009), damage mechanics (Verhoosel et al., 2010a), cohesive zone simulations (Verhoosel et al., 2010b), heat transfer (Anders, Weinberg and Reichart, 2012), large deformation (Benson et al., 2011), electromagnetic (Buffa, Sangalli and Vazquez, 2010), strain localization (Elguedj, Rethore and Buteri, 2011), contact mechanics (Lu, 2011; Temizer, Wriggers and Hughes, 2011), topology optimization (Hassani, Khanzadi and Tavakkoli, 2012) and crack propagation (Verhoosel et al., 2010b, Benson et al., 2010b; De Luycker et al., 2011; Haasemann et al., 2011; Ghorashi, Valizadeh and Mohammadi, 2012).

The first attempt at enhancing IGA for crack problems was reported by Verhoosel et al. (2010b) and followed by Benson et al. (2010c), De Luycker et al. (2011) and Haasemann et al. (2011). Recently, a full combination of XFEM and IGA methodologies has been developed for general mixed mode crack propagation problems by the introduction of extended isogeometric analysis (XIGA) by Ghorashi, Valizadeh and Mohammadi (2012). XIGA uses the superior concepts of XFEM to extrinsically enrich the versatile IGA control points with Heaviside and crack-tip enrichment functions.

1.5.7 Multiscale Analysis