EXTENDED FINITE ELEMENT METHOD

for Fracture Analysis of Structures

Soheil MohammadiSchool of Civil Engineering

University of TehranTehran, Iran

EXTENDED FINITE ELEMENT METHOD

for Fracture Analysis of Structures

Soheil MohammadiSchool of Civil Engineering

University of TehranTehran, Iran

C© 2008 by Soheil MohammadiPublished by Blackwell Publishing Ltd

Editorial offices:Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK

Tel: +44 (0) 1865 776868Blackwell Publishing Inc., 350 Main Street, Malden, MA 02148-5020, USA

Tel: +1 781 388 8250Blackwell Publishing Asia Pty Ltd, 550 Swanston Street, Carlton, Victoria 3053, Australia

Tel: +61 (0)3 8359 1011

The right of the Author to be identified as the Author of this Work has been asserted inaccordance with the UK Copyright, Designs, and Patents Act 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, 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 Act1988, without the prior permission of the publisher.

Designations used by companies to distinguish their products are often claimed as trade-marks. All brand names and product names used in this book are trade names, servicemarks, trademarks or registered trademarks of their respective owners. The Publisher is notassociated with any product or vendor mentioned in this book.

This publication is designed to provide accurate and authoritative information in regard tothe subject matter covered. It is sold on the understanding that the Publisher is not engaged inrendering professional services. If professional advice or other expert assistance is required,the services of a competent professional should be sought.

First published 2008 by Blackwell Publishing Ltd

ISBN: 978-1-4051-7060-4

Library of Congress Cataloging-in-Publication Data

Mohammadi, Soheil.Extended finite element method for fracture analysis of structures / Soheil Mohammadi.

p. cm.Includes bibliographical references and index.ISBN-13: 978-1-4051-7060-4 (hardback : alk. paper)ISBN-10: 1-4051-7060-3 (hardback : alk. paper)1. Fracture mechanics. 2. Finite element method. I. Title.

TA409.M65 2007624.1′76–dc22

2007018717

A catalogue record for this title is available from the British Library

Typeset by Soheil MohammadiPrinted and bound in Singaporeby Utopia Press Pte Ltd

The publisherís policy is to use permanent paper from mills that operate a sustainableforestry policy, and which has been manufactured from pulp processed using acid-free andelementary chlorine-free practices. Furthermore, the publisher ensures that the text paperand cover board used have met acceptable environmental accreditation standards.

For further information on Blackwell Publishing, visit our website:www.blackwellpublishing.com/construction

Contents

Dedication viiiPreface ixNomenclature xi

Chapter 1 Introduction1.1 ANALYSIS OF STRUCTURES 11.2 ANALYSIS OF DISCONTINUITIES 21.3 FRACTURE MECHANICS 31.4 CRACK MODELLING 3

1.4.1 Local and non-local models 41.4.2 Smeared crack model 41.4.3 Discrete inter-element crack 61.4.4 Discrete cracked element 61.4.5 Singular elements 61.4.6 Enriched elements 7

1.5 ALTERNATIVE TECHNIQUES 71.6 A REVIEW OF XFEM APPLICATIONS 7

1.6.1 General aspects of XFEM 71.6.2 Localisation and fracture 81.6.3 Composites 91.6.4 Contact 91.6.5 Dynamics 91.6.6 Large deformation/shells 101.6.7 Multiscale 101.6.8 Multiphase/solidification 10

1.7 SCOPE OF THE BOOK 11

Chapter 2 Fracture Mechanics, a Review2.1 INTRODUCTION 132.2 BASICS OF ELASTICITY 14

2.2.1 Stress–strain relations 142.2.2 Airy stress function 172.2.3 Complex stress functions 18

2.3 BASICS OF LEFM 192.3.1 Fracture mechanics 192.3.2 Circular hole 202.3.3 Elliptical hole 212.3.4 Westergaard analysis of a sharp crack 22

iv Contents

2.4 STRESS INTENSITY FACTOR, K 242.4.1 Definition of the stress intensity factor 242.4.2 Examples of stress intensity factors for LEFM 282.4.3 Griffith theories of strength and energy 292.4.4 Brittle material 302.4.5 Quasi-brittle material 322.4.6 Crack stability 322.4.7 Fixed grip versus fixed load 332.4.8 Mixed mode crack propagation 34

2.5 SOLUTION PROCEDURES FOR K AND G 382.5.1 Displacement extrapolation/correlation method 382.5.2 Mode I energy release rate 392.5.3 Mode I stiffness derivative/virtual crack model 402.5.4 Two virtual crack extensions for mixed mode cases 412.5.5 Single virtual crack extension based on displacement decomposition 422.5.6 Quarter point singular elements 43

2.6 ELASTOPLASTIC FRACTURE MECHANICS (EPFM) 472.6.1 Plastic zone 472.6.2 Crack tip opening displacements (CTOD) 502.6.3 J integral 512.6.4 Plastic crack tip fields 542.6.5 Generalisation of J 55

2.7 NUMERICAL METHODS BASED ON THE J INTEGRAL 562.7.1 Nodal solution 562.7.2 General finite element solution 572.7.3 Equivalent domain integral (EDI) method 592.7.4 Interaction integral method 59

Chapter 3 Extended Finite Element Method for Isotropic Problems3.1 INTRODUCTION 613.2 A REVIEW OF XFEM DEVELOPMENT 613.3 BASICS OF FEM 65

3.3.1 Isoparametric finite elements, a short review 653.3.2 Finite element solutions for fracture mechanics 67

3.4 PARTITION OF UNITY 683.5 ENRICHMENT 69

3.5.1 Intrinsic enrichment 703.5.2 Extrinsic enrichment 713.5.3 Partition of unity finite element method 723.5.4 Generalised finite element method 733.5.5 Extended finite element method 733.5.6 Hp-clouds enrichment 733.5.7 Generalisation of the PU enrichment 743.5.8 Transition from standard to enriched approximation 74

3.6 ISOTROPIC XFEM 763.6.1 Basic XFEM approximation 763.6.2 Signed distance function 783.6.3 Modelling strong discontinuous fields 783.6.4 Modelling weak discontinuous fields 85

Contents v

3.6.5 Plastic enrichment 863.6.6 Selection of nodes for discontinuity enrichment 873.6.7 Modelling the crack 90

3.7 DISCRETIZATION AND INTEGRATION 903.7.1 Governing equation 903.7.2 XFEM discretization 913.7.3 Element partitioning and numerical integration 953.7.4 Crack intersection 96

3.8 TRACKING MOVING BOUNDARIES 973.8.1 Level set method 983.8.2 Fast marching method 1053.8.3 Ordered upwind method 108

3.9 NUMERICAL SIMULATIONS 1093.9.1 A tensile plate with a central crack 1093.9.2 Double edge cracks 1113.9.3 Double internal collinear cracks 1123.9.4 A central crack in an infinite plate 1143.9.5 An edge crack in a finite plate 115

Chapter 4 XFEM for Orthotropic Problems4.1 INTRODUCTION 1174.2 ANISOTROPIC ELASTICITY 118

4.2.1 Elasticity solution 1184.2.2 Anisotropic stress functions 1204.2.3 Orthotropic mixed mode problems 1214.2.4 Energy release rate and stress intensity factor for anisotropic

materials 1224.2.5 Anisotropic singular elements 123

4.3 ANALYTICAL SOLUTIONS FOR NEAR CRACK TIP 1244.3.1 Near crack tip displacement field (class I) 1244.3.2 Near crack tip displacement field (class II) 1284.3.3 Unified near crack tip displacement field (both classes) 131

4.4 ANISOTROPIC XFEM 1344.4.1 Governing equation 1344.4.2 XFEM discretization 1354.4.3 SIF calculations 137

4.5 NUMERICAL SIMULATIONS 1404.5.1 Plate with a crack parallel to material axis of orthotropy 1424.5.2 Edge crack with several orientations of the axes of orthotropy 1444.5.3 Single edge notched tensile specimen with crack inclination 1474.5.4 Central slanted crack 1504.5.5 An inclined centre crack in a disk subjected to point loads 1564.5.6 A crack between orthotropic and isotropic materials subjected to

tensile tractions 160

Chapter 5 XFEM for Cohesive Cracks5.1 INTRODUCTION 1635.2 COHESIVE CRACKS 164

5.2.1 Cohesive crack models 166

vi Contents

5.2.2 Numerical models for cohesive cracks 1685.2.3 Crack propagation criteria 1725.2.4 Snap-back behaviour 1735.2.5 Griffith criterion for cohesive crack 1745.2.6 Cohesive crack model 175

5.3 XFEM FOR COHESIVE CRACKS 1765.3.1 Enrichment functions 1765.3.2 Governing equations 1795.3.3 XFEM discretization 181

5.4 NUMERICAL SIMULATIONS 1835.4.1 Mixed mode bending beam 1835.4.2 Four point bending beam 1845.4.3 Double cantilever beam 187

Chapter 6 New Frontiers6.1 INTRODUCTION 1896.2 INTERFACE CRACKS 190

6.2.1 Elasticity solution for isotropic bimaterial interface 1906.2.2 Stability of interface cracks 1916.2.3 XFEM approximation for interface cracks 193

6.3 CONTACT 1946.3.1 Numerical models for a contact problem 1946.3.2 XFEM modelling of a contact problem 195

6.4 DYNAMIC FRACTURE 1986.4.1 Dynamic crack propagation by XFEM 1986.4.2 Dynamic LEFM 1996.4.3 Dynamic orthotropic LEFM 2006.4.4 Basic formulation of dynamic XFEM 2026.4.5 XFEM discretization 2036.4.6 Time integration 2056.4.7 Time finite element method 2066.4.8 Time extended finite element method 207

6.5 MULTISCALE XFEM 2076.5.1 Basic formulation 2076.5.2 The zoom technique 2086.5.3 Homogenisation based techniques 2096.5.4 XFEM discretization 210

6.6 MULTIPHASE XFEM 2116.6.1 Basic formulation 2116.6.2 XFEM approximation 2126.6.3 Two-phase fluid flow 2146.6.4 XFEM approximation 215

Chapter 7 XFEM Flow7.1 INTRODUCTION 2197.2 AVAILABLE OPEN-SOURCE XFEM 2207.3. FINITE ELEMENT ANALYSIS 220

7.3.1 Defining the model 220

Contents vii

7.3.2 Creating the finite element mesh 2207.3.3 Linear elastic analysis 2217.3.4 Large deformation 2217.3.5 Nonlinear (elastoplastic) analysis 2227.3.6 Material constitutive matrix 223

7.4 XFEM 2237.4.1 Front tracking 2237.4.2 Enrichment detection 2257.4.3 Enrichment functions 2267.4.4 Ramp (transition) functions 2297.4.5 Evaluation of the B matrix 229

7.5 NUMERICAL INTEGRATION 2297.5.1 Sub-quads 2307.5.2 Sub-triangles 230

7.6 SOLVER 2317.6.1 XFEM degrees of freedom 2317.6.2 Time integration 2317.6.3 Simultaneous equations solver 2327.6.4 Crack length control 232

7.7 POST-PROCESSING 2337.7.1 Stress intensity factor 2337.7.2 Crack growth 2337.7.3 Other applications 234

7.8 CONFIGURATION UPDATE 234

References 235Index 249

To Mansoureh

Preface

‘I am always obliged to a person who has taught me a single word.’

Progressive failure/fracture analysis of structures has been an active research topic forthe past two decades. Historically, it has been addressed either within the frameworkof continuum computational plasticity and damage mechanics, or the discontinuousapproach of fracture mechanics. The present form of linear elastic fracture mechanics(LEFM), with its roots a century old has since been successfully applied to variousclassical crack and defect problems. Nevertheless, it remains relatively limited to simplegeometries and loading conditions, unless coupled with a powerful numerical tool suchas the finite element method and meshless approaches.

The finite element method (FEM) has undoubtedly become the most popular andpowerful analytical tool for studying a wide range of engineering and physical prob-lems. Several general purpose finite element codes are now available and concepts ofFEM are usually offered by all engineering departments in the form of postgraduateand even undergraduate courses. Singular elements, adaptive finite element procedures,and combined finite/discrete element methodologies have substantially contributed tothe development and accuracy of fracture analysis of structures. Despite all achieve-ments, the continuum basis of FEM remained a source of relative disadvantage fordiscontinuous fracture mechanics. After a few decades, a major breakthrough seemsto have been made by the fundamental idea of partition of unity and in the form of theeXtended Finite Element Method (XFEM).

This book has been prepared primarily to introduce the concepts of the newlydeveloped extended finite element method for fracture analysis of structures. An at-tempt has also been made to discuss the essential features of XFEM for other relatedengineering applications. The book can be divided into four parts. The first part is ded-icated to the basic concepts and fundamental formulations of fracture mechanics. Itcovers discussions on classical problems of LEFM and their extension to elastoplasticfracture mechanics (EPFM). Issues related to the standard finite element modellingof fracture mechanics and the basics of popular singular finite elements are reviewedbriefly.

The second part, which constitutes most of the book, is devoted to a detailed dis-cussion on various aspects of XFEM. It begins by discussing fundamentals of partitionof unity and basics of XFEM formulation in Chapter 3. Effects of various enrichmentfunctions, such as crack tip, Heaviside and weak discontinuity enrichment functions arealso investigated. Two commonly used level set and fast marching methods for track-ing moving boundaries are explained before the chapter is concluded by examining anumber of classical problems of fracture mechanics. The next chapter deals with theorthotropic fracture mechanics as an extension of XFEM for ever growing applications

x Preface

of composite materials. A different set of enrichment functions for orthotropic mediais presented, followed by a number of simulations of benchmark orthotropic problems.Chapter 5, devoted to simulation of cohesive cracks by XFEM, provides theoreticalbases for cohesive crack models in fracture mechanics, classical FEM and XFEM.The snap-back response and the concept of critical crack path are studied by solving anumber of classical cohesive crack problems.

The third part of the book (Chapter 6) provides basic information on new frontiersof application of XFEM. It begins with discussions on interface cracking, which includeclassical solutions from fracture mechanics and XFEM approximation. Application ofXFEM for solving contact problems is explained and numerical issues are addressed.The important subject of dynamic fracture is then discussed by introducing classicalformulations of fracture mechanics and the recently developed idea of time–spacediscretization by XFEM. New extensions of XFEM for very complex applications ofmultiscale and multiphase problems are explained briefly.

The final chapter explains a number of simple instructions, step-by-step proce-dures and algorithms for implementing an efficient XFEM. These simple guidelines, incombination with freely available XFEM source codes, can be used to further advancethe existing XFEM capabilities.

This book is the result of an infinite number of brilliant research works in thefield of computational mechanics for many years all over the world. I have tried toappropriately acknowledge the achievements of corresponding authors within the text,relevant figures, tables and formulae. I am much indebted to their outstanding researchworks and any unintentional shortcoming in sufficiently acknowledging them is sin-cerely regretted. Perhaps such a title should have become available earlier by one ofthe pioneers of the method, i.e. Professor T. Belytschko, a shining star in the universeof computational mechanics, Dr J. Dolbow, Dr N. Moes, Dr N. Sukumar and possiblyothers who introduced, contributed and developed most of the techniques.

I would like to extend my acknowledgement to Blackwell Publishing Limited,for facilitating the publication of the first book on XFEM; in particular N. Warnock-Smith, J. Burden, L. Alexander, A. Cohen and A. Hallam for helping me throughoutthe work. Also, I would like to express my sincere gratitude to my long-time friend,Professor A.R. Khoei, with whom I have had many discussions on various subjects ofcomputational mechanics, including XFEM. Also my special thanks go to my students:Mr A. Asadpoure, to whom I owe most of Chapter 4, Mr S.H. Ebrahimi for solvingisotropic examples in Chapter 3 and Mr A. Forghani for providing some of the resultsin Chapter 5.

This book has been completed on the eve of the new Persian year; a ‘temporalinterface’ between winter and spring, and an indication of the beginning of a bloomingseason for XFEM, I hope.

Finally, I would like to express my gratitude to my family for their love, under-standing and never-ending support. I have spent many hours on writing this book; hoursthat could have been devoted to my wife and little Sogol: the spring flowers that inspirethe life.

Soheil MohammadiTehran, IranSpring 2007

Nomenclature

α Curvilinear coordinateαc Load factor for cohesionα f , αs Thermal diffusivity of fluid and solid phasesβ Curvilinear coordinateγs Surface energy densityγ e

s , γps Elastic and plastic surface energies

γxy Engineering shear strainδ Plastic crack tip zoneδ Variation of a functionδ(ξ ) Dirac delta functionδi j Kronecker delta functionεε Strain tensorεε f , εεc Strain field at fine and coarse scalesεij Strain componentsεij Dimensionless angular geometric functionεaux

ij Auxiliary strain components

εv Kinetic mobility coefficientεyld Yield strainη Local curvilinear (mapping) coordinate systemθ Crack propagation angle with respect to initial crackθ Angular polar coordinateκ, κ ′ Material parametersλ Lame modulusλ Eigenvalue of the characteristic equationμ Shear modulusν, νij Isotropic and orthotropic Poisson’s ratiosξ Local curvilinear (mapping) coordinate systemξ(x) Distance functionρ Radius of curvatureρ Densityρ f , ρc Density of fine and coarse scalesρint Curvature of the propagating interfaceσ Stress tensorσ f ,σc Stress field at fine and coarse scalesσg Stress tensor at a Gauss point

σtipt Normal tensile stress at crack tip

σ0 Applied normal traction

xii Nomenclature

σcr Critical stress for crackingσij Stress componentsσij Dimensionless angular geometric functionσ aux

ij Auxiliary stress components

σn Stress component normal to an interfaceσn Stress component at time step nσyld Yield stressτ Deviatoric stressτ0 Applied tangential tractionτc Cohesive shear tractionτn Time functionsτn Deviatoric stress tensor at time step nφ(x) Level set functionφ(z) Complex stress functionφs(z) Stress function for shear problemϕ Angle of orthotropic axesϕ Phase angle for interface fractureχ(x) Enrichment function for weak discontinuitiesχ(z) Stress functionψ (x) Enrichment functionψ (z) Stress functionω Oscillation index� Boundary�c Crack boundary�t Traction (natural) boundary�u Displacement (essential) boundary� Finite variation of a function� Coefficient matrixΞ Homogenisation/average operator� Potential energy� j (x) Moving least squares shape functions�(x) Stress function� Domain� f , �c Fine and course scale domains� f , �s Fluid and solid domains�pu Domain associated with the partition of unity

a Crack length/half lengtha Semi-major axis of ellipseab, af Backward and forward indexes in fast marching methodah Heaviside enrichment degrees of freedomai Enrichment degrees of freedomak Enrichment degrees of freedomA∗ Area associated with the domain J integralb Width of a plateb Semi-minor axis of ellipsebi Crack tip enrichment degrees of freedom

Nomenclature xiii

B Matrix of derivatives of shape functionsBh Matrix of derivatives of final shape functionsBc B matrix for coarse scaleB f B matrix for fine scaleBr

i Strain–displacement matrix (derivatives of shape functions)Bu

i Strain–displacement matrix (derivatives of shape functions)Ba

i Matrix of derivatives of enrichment (Heaviside) of shape functionsBb

i Matrix of derivatives of enrichment (crack tip) of shape functionsc Constant parameterc Size of crack tip contour for J integralcij Material constantscR Rayleigh speedc f , cs Specific heat for fluid and solid phasesC Material constitutive matrixd Distanced/dt Time derivativeD Material modulus matrixDc, D f Material modulus in coarse and fine scalesDloc Localisation modulusD/Dt Material time derivativeDb

x , Dfx Backward and forward finite difference approximations

E, Ei Isotropic and orthotropic Young’s modulusE ′ Material parameterft Uniaxial tensile strengthf (r ) Radial functionf Nodal force vectorf r

i Nodal force components (classic and enriched)

f b Body force vectorf t External traction vectorf c Cohesive crack traction vectorf coh Cohesive nodal force vectorf ext External force vectorf int

u Internal nodal force vector due to external loadingf int

a Internal nodal force vector due to cohesive forceFi

l (x) Crack tip enrichment functionsg Applied gravitational body forceg(θ ) Angular function for a crack tip kink problemg j (θ ) Orthotropic crack tip enrichment functionsG Shear modulusG Fracture energy release rateG1, G2 Mode I and II fracture energy release rates

GdynI Dynamic mode I fracture energy release

H (ξ) Heaviside functionHl Latent heati Complex number, i2 = −1J Jacobian matrixJ J integral