EXTENDED FINITE ELEMENT METHOD

Transcription

1 EXTENDED FINITE ELEMENT METHOD for Fracture Analysis of Structures Soheil Mohammadi School of Civil Engineering University of Tehran Tehran, Iran

2 C 2008 by Soheil Mohammadi Published by Blackwell Publishing Ltd Editorial offices: Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK Tel: +44 (0) Blackwell Publishing Inc., 350 Main Street, Malden, MA , USA Tel: Blackwell Publishing Asia Pty Ltd, 550 Swanston Street, Carlton, Victoria 3053, Australia Tel: +61 (0) The right of the Author to be identified as the Author of this Work has been asserted in accordance with the UK Copyright, Designs, and Patents Act 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. 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. First published 2008 by Blackwell Publishing Ltd ISBN: 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: (hardback : alk. paper) ISBN-10: (hardback : alk. paper) 1. Fracture mechanics. 2. Finite element method. I. Title. TA409.M dc A catalogue record for this title is available from the British Library Typeset by Soheil Mohammadi Printed and bound in Singapore by Utopia Press Pte Ltd The publisherís policy is to use permanent paper from mills that operate a sustainable forestry policy, and which has been manufactured from pulp processed using acid-free and elementary chlorine-free practices. Furthermore, the publisher ensures that the text paper and cover board used have met acceptable environmental accreditation standards. For further information on Blackwell Publishing, visit our website:

3 Contents Dedication Preface Nomenclature viii ix xi Chapter 1 Introduction 1.1 ANALYSIS OF STRUCTURES ANALYSIS OF DISCONTINUITIES FRACTURE MECHANICS CRACK MODELLING Local and non-local models Smeared crack model Discrete inter-element crack Discrete cracked element Singular elements Enriched elements ALTERNATIVE TECHNIQUES A REVIEW OF XFEM APPLICATIONS General aspects of XFEM Localisation and fracture Composites Contact Dynamics Large deformation/shells Multiscale Multiphase/solidification SCOPE OF THE BOOK 11 Chapter 2 Fracture Mechanics, a Review 2.1 INTRODUCTION BASICS OF ELASTICITY Stress strain relations Airy stress function Complex stress functions BASICS OF LEFM Fracture mechanics Circular hole Elliptical hole Westergaard analysis of a sharp crack 22

4 iv Contents 2.4 STRESS INTENSITY FACTOR, K Definition of the stress intensity factor Examples of stress intensity factors for LEFM Griffith theories of strength and energy Brittle material Quasi-brittle material Crack stability Fixed grip versus fixed load Mixed mode crack propagation SOLUTION PROCEDURES FOR K AND G Displacement extrapolation/correlation method Mode I energy release rate Mode I stiffness derivative/virtual crack model Two virtual crack extensions for mixed mode cases Single virtual crack extension based on displacement decomposition Quarter point singular elements ELASTOPLASTIC FRACTURE MECHANICS (EPFM) Plastic zone Crack tip opening displacements (CTOD) J integral Plastic crack tip fields Generalisation of J NUMERICAL METHODS BASED ON THE J INTEGRAL Nodal solution General finite element solution Equivalent domain integral (EDI) method Interaction integral method 59 Chapter 3 Extended Finite Element Method for Isotropic Problems 3.1 INTRODUCTION A REVIEW OF XFEM DEVELOPMENT BASICS OF FEM Isoparametric finite elements, a short review Finite element solutions for fracture mechanics PARTITION OF UNITY ENRICHMENT Intrinsic enrichment Extrinsic enrichment Partition of unity finite element method Generalised finite element method Extended finite element method Hp-clouds enrichment Generalisation of the PU enrichment Transition from standard to enriched approximation ISOTROPIC XFEM Basic XFEM approximation Signed distance function Modelling strong discontinuous fields Modelling weak discontinuous fields 85

5 Contents v Plastic enrichment Selection of nodes for discontinuity enrichment Modelling the crack DISCRETIZATION AND INTEGRATION Governing equation XFEM discretization Element partitioning and numerical integration Crack intersection TRACKING MOVING BOUNDARIES Level set method Fast marching method Ordered upwind method NUMERICAL SIMULATIONS A tensile plate with a central crack Double edge cracks Double internal collinear cracks A central crack in an infinite plate An edge crack in a finite plate 115 Chapter 4 XFEM for Orthotropic Problems 4.1 INTRODUCTION ANISOTROPIC ELASTICITY Elasticity solution Anisotropic stress functions Orthotropic mixed mode problems Energy release rate and stress intensity factor for anisotropic materials Anisotropic singular elements ANALYTICAL SOLUTIONS FOR NEAR CRACK TIP Near crack tip displacement field (class I) Near crack tip displacement field (class II) Unified near crack tip displacement field (both classes) ANISOTROPIC XFEM Governing equation XFEM discretization SIF calculations NUMERICAL SIMULATIONS Plate with a crack parallel to material axis of orthotropy Edge crack with several orientations of the axes of orthotropy Single edge notched tensile specimen with crack inclination Central slanted crack An inclined centre crack in a disk subjected to point loads A crack between orthotropic and isotropic materials subjected to tensile tractions 160 Chapter 5 XFEM for Cohesive Cracks 5.1 INTRODUCTION COHESIVE CRACKS Cohesive crack models 166

6 vi Contents Numerical models for cohesive cracks Crack propagation criteria Snap-back behaviour Griffith criterion for cohesive crack Cohesive crack model XFEM FOR COHESIVE CRACKS Enrichment functions Governing equations XFEM discretization NUMERICAL SIMULATIONS Mixed mode bending beam Four point bending beam Double cantilever beam 187 Chapter 6 New Frontiers 6.1 INTRODUCTION INTERFACE CRACKS Elasticity solution for isotropic bimaterial interface Stability of interface cracks XFEM approximation for interface cracks CONTACT Numerical models for a contact problem XFEM modelling of a contact problem DYNAMIC FRACTURE Dynamic crack propagation by XFEM Dynamic LEFM Dynamic orthotropic LEFM Basic formulation of dynamic XFEM XFEM discretization Time integration Time finite element method Time extended finite element method MULTISCALE XFEM Basic formulation The zoom technique Homogenisation based techniques XFEM discretization MULTIPHASE XFEM Basic formulation XFEM approximation Two-phase fluid flow XFEM approximation 215 Chapter 7 XFEM Flow 7.1 INTRODUCTION AVAILABLE OPEN-SOURCE XFEM FINITE ELEMENT ANALYSIS Defining the model 220

7 Contents vii Creating the finite element mesh Linear elastic analysis Large deformation Nonlinear (elastoplastic) analysis Material constitutive matrix XFEM Front tracking Enrichment detection Enrichment functions Ramp (transition) functions Evaluation of the B matrix NUMERICAL INTEGRATION Sub-quads Sub-triangles SOLVER XFEM degrees of freedom Time integration Simultaneous equations solver Crack length control POST-PROCESSING Stress intensity factor Crack growth Other applications CONFIGURATION UPDATE 234 References 235 Index 249

8 To Mansoureh

9 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 for the past two decades. Historically, it has been addressed either within the framework of continuum computational plasticity and damage mechanics, or the discontinuous approach of fracture mechanics. The present form of linear elastic fracture mechanics (LEFM), with its roots a century old has since been successfully applied to various classical crack and defect problems. Nevertheless, it remains relatively limited to simple geometries and loading conditions, unless coupled with a powerful numerical tool such as the finite element method and meshless approaches. The finite element method (FEM) has undoubtedly become the most popular and powerful analytical tool for studying a wide range of engineering and physical problems. Several general purpose finite element codes are now available and concepts of FEM are usually offered by all engineering departments in the form of postgraduate and even undergraduate courses. Singular elements, adaptive finite element procedures, and combined finite/discrete element methodologies have substantially contributed to the development and accuracy of fracture analysis of structures. Despite all achievements, the continuum basis of FEM remained a source of relative disadvantage for discontinuous fracture mechanics. After a few decades, a major breakthrough seems to have been made by the fundamental idea of partition of unity and in the form of the extended Finite Element Method (XFEM). This book has been prepared primarily to introduce the concepts of the newly developed extended finite element method for fracture analysis of structures. An attempt has also been made to discuss the essential features of XFEM for other related engineering applications. The book can be divided into four parts. The first part is dedicated to the basic concepts and fundamental formulations of fracture mechanics. It covers discussions on classical problems of LEFM and their extension to elastoplastic fracture mechanics (EPFM). Issues related to the standard finite element modelling of fracture mechanics and the basics of popular singular finite elements are reviewed briefly. The second part, which constitutes most of the book, is devoted to a detailed discussion on various aspects of XFEM. It begins by discussing fundamentals of partition of unity and basics of XFEM formulation in Chapter 3. Effects of various enrichment functions, such as crack tip, Heaviside and weak discontinuity enrichment functions are also investigated. Two commonly used level set and fast marching methods for tracking moving boundaries are explained before the chapter is concluded by examining a number of classical problems of fracture mechanics. The next chapter deals with the orthotropic fracture mechanics as an extension of XFEM for ever growing applications

10 x Preface of composite materials. A different set of enrichment functions for orthotropic media is presented, followed by a number of simulations of benchmark orthotropic problems. Chapter 5, devoted to simulation of cohesive cracks by XFEM, provides theoretical bases 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 a number of classical cohesive crack problems. The third part of the book (Chapter 6) provides basic information on new frontiers of application of XFEM. It begins with discussions on interface cracking, which include classical solutions from fracture mechanics and XFEM approximation. Application of XFEM for solving contact problems is explained and numerical issues are addressed. The important subject of dynamic fracture is then discussed by introducing classical formulations of fracture mechanics and the recently developed idea of time space discretization by XFEM. New extensions of XFEM for very complex applications of multiscale and multiphase problems are explained briefly. The final chapter explains a number of simple instructions, step-by-step procedures and algorithms for implementing an efficient XFEM. These simple guidelines, in combination with freely available XFEM source codes, can be used to further advance the existing XFEM capabilities. This book is the result of an infinite number of brilliant research works in the field of computational mechanics for many years all over the world. I have tried to appropriately acknowledge the achievements of corresponding authors within the text, relevant figures, tables and formulae. I am much indebted to their outstanding research works and any unintentional shortcoming in sufficiently acknowledging them is sincerely regretted. Perhaps such a title should have become available earlier by one of the pioneers of the method, i.e. Professor T. Belytschko, a shining star in the universe of computational mechanics, Dr J. Dolbow, Dr N. Moës, Dr N. Sukumar and possibly others 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 throughout the 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 of computational 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 solving isotropic examples in Chapter 3 and Mr A. Forghani for providing some of the results in Chapter 5. This book has been completed on the eve of the new Persian year; a temporal interface between winter and spring, and an indication of the beginning of a blooming season for XFEM, I hope. Finally, I would like to express my gratitude to my family for their love, understanding and never-ending support. I have spent many hours on writing this book; hours that could have been devoted to my wife and little Sogol: the spring flowers that inspire the life. Soheil Mohammadi Tehran, Iran Spring 2007

11 Nomenclature α Curvilinear coordinate α c Load factor for cohesion α f,α s Thermal diffusivity of fluid and solid phases β Curvilinear coordinate γ s Surface energy density γs e,γp s Elastic and plastic surface energies γ xy Engineering shear strain δ Plastic crack tip zone δ Variation of a function δ(ξ) Dirac delta function δ ij Kronecker delta function ε Strain tensor ε f, ε c Strain field at fine and coarse scales ε ij Strain components ε ij Dimensionless angular geometric function εij aux Auxiliary strain components ε v Kinetic mobility coefficient ε yld Yield strain η Local curvilinear (mapping) coordinate system θ Crack propagation angle with respect to initial crack θ Angular polar coordinate κ, κ Material parameters λ Lame modulus λ Eigenvalue of the characteristic equation μ Shear modulus ν, ν ij Isotropic and orthotropic Poisson s ratios ξ Local curvilinear (mapping) coordinate system ξ(x) Distance function ρ Radius of curvature ρ Density ρ f,ρ c Density of fine and coarse scales ρ int Curvature of the propagating interface σ Stress tensor σ f, σ c Stress field at fine and coarse scales σ g Stress tensor at a Gauss point σ tip t Normal tensile stress at crack tip σ 0 Applied normal traction

12 xii Nomenclature σ cr σ ij σ ij σij aux σ n σ n σ yld τ τ 0 τ c τ n τ n φ(x) φ(z) φ s (z) ϕ ϕ χ(x) χ(z) ψ(x) ψ(z) ω Ɣ Ɣ c Ɣ t Ɣ u Ξ j (x) (x) f, c f, s pu a a a b, a f a h a i a k A b b b i Critical stress for cracking Stress components Dimensionless angular geometric function Auxiliary stress components Stress component normal to an interface Stress component at time step n Yield stress Deviatoric stress Applied tangential traction Cohesive shear traction Time functions Deviatoric stress tensor at time step n Level set function Complex stress function Stress function for shear problem Angle of orthotropic axes Phase angle for interface fracture Enrichment function for weak discontinuities Stress function Enrichment function Stress function Oscillation index Boundary Crack boundary Traction (natural) boundary Displacement (essential) boundary Finite variation of a function Coefficient matrix Homogenisation/average operator Potential energy Moving least squares shape functions Stress function Domain Fine and course scale domains Fluid and solid domains Domain associated with the partition of unity Crack length/half length Semi-major axis of ellipse Backward and forward indexes in fast marching method Heaviside enrichment degrees of freedom Enrichment degrees of freedom Enrichment degrees of freedom Area associated with the domain J integral Width of a plate Semi-minor axis of ellipse Crack tip enrichment degrees of freedom

13 Nomenclature xiii B Matrix of derivatives of shape functions B h Matrix of derivatives of final shape functions B c B matrix for coarse scale B f B matrix for fine scale B r i Strain displacement matrix (derivatives of shape functions) Bi u Strain displacement matrix (derivatives of shape functions) Bi a Matrix of derivatives of enrichment (Heaviside) of shape functions Bi b Matrix of derivatives of enrichment (crack tip) of shape functions c Constant parameter c Size of crack tip contour for J integral c ij Material constants c R Rayleigh speed c f, c s Specific heat for fluid and solid phases C Material constitutive matrix d Distance d/dt Time derivative D Material modulus matrix D c, D f Material modulus in coarse and fine scales D loc Localisation modulus D/Dt Material time derivative Dx b, Df x Backward and forward finite difference approximations E, E i Isotropic and orthotropic Young s modulus E Material parameter f t Uniaxial tensile strength f (r) Radial function f Nodal force vector f r i Nodal force components (classic and enriched) f b Body force vector f t External traction vector f c Cohesive crack traction vector f coh Cohesive nodal force vector f ext External force vector f u int Internal nodal force vector due to external loading f a int Internal nodal force vector due to cohesive force Fl i Crack tip enrichment functions g Applied gravitational body force g(θ) Angular function for a crack tip kink problem g j (θ) Orthotropic crack tip enrichment functions G Shear modulus G Fracture energy release rate G 1, G 2 Mode I and II fracture energy release rates G dyn I Dynamic mode I fracture energy release H(ξ) Heaviside function H l Latent heat i Complex number, i 2 = 1 J Jacobian matrix J J integral

14 xiv Nomenclature J act Actual J integral J aux Auxiliary J integral J k Mode k contour integral J k 0 Dimensionless constant for the power hardening law k 0, k 1, k 2, k 3, k 4 Constant coefficients k i Conductivity coefficient for phase i k s, k f Thermal conductivity for solid and fluid phases k n, k t Normal/tangential interface properties K Stiffness matrix K hom Homogenised stiffness matrix K rs ij Stiffness matrix components K Stress intensity factor K C Critical stress intensity factor K eq Equivalent mixed mode stress intensity factor K I, K II, K III Mode I, II and III stress intensity factors K I, K II Normalized mode I and mode II stress intensity factors KI aux, KII aux Auxiliary mode I and mode II stress intensity factors K Ic, K IIc Critical mode I and mode II stress intensity factors KI cohesion Cohesive mode I stress intensity factor KI crack Crack mode I stress intensity factor K dyn I Dynamic mode I stress intensity factor l e Characteristic length l c Characteristic length for crack propagaion m Number of enrichment functions mt i Number of nodes to be enriched by crack tip enrichment functions mf Number of crack tip enrichment functions M j Mach number M Interaction integral M 0 Total mass M i Lumped mass component M ij Mass matrix component n Power number for the plastic model (Section 2.6.4) ng Number of Gauss points n Number of nodes within each moving least squares support domain np Number of independent domains of partition of unity n n Number of nodes in a finite element n Normal vector n int Normal vector to an internal interface N j Matrix of shape functions N j Shape function N j New set of generalised finite element method shape functions p(x) Basis function p Hydrostatic pressure p Predefined hydrostatic pressure P i Loading condition i q Arbitrary smoothing function q Heat flux

15 Nomenclature xv q i Q Q ij r r g r p R s f, s s S a S c S d t t t int T T f, T s T i (t) T m u û u ū u ü ui aux u coh u enr u FE u h (x) ū j ũ j u p u x, u y U 1, U 2 U k U s Us e, U s p U Ɣ v v v n v 1,v 2 V(t) w w c W W aux Nodal values of the arbitrary smoothing function Input heat to system Matrix of homogenous anisotropic solids Radial distance/coordinate Radial distance of a Gauss point from crack tip Crack tip plastic zone Ramp function Heat source for fluid and solid phases Set of accepted nodes Set of candidate nodes Set of distant nodes Time Traction Surface traction along internal boundary Temperature Temperature of fluid and solid phases Time shape functions Melting/fusion temperature Displacement vector Local symmetric displacement vector Velocity vector Prescribed displacement Prescribed velocity Acceleration vector Auxiliary displacement field Displacement field obtained from crack surface tractions Enriched displacement field Classical finite element displacement field Approximated displacement field Nodal displacement vector Transformed displacement Periodic displacement xand y displacement component Symmetric and antisymmetric crack tip displacements Kinetic energy Strain energy Elastic and plastic strain energies Surface energy Velocity vector Prescribed velocity Normal interface speed Longitudinal and shear wave velocities Vector of approximated velocity degrees of freedom Crack opening Critical crack opening External work Auxiliary work

16 xvi Nomenclature W coh W ext W g Wg r W int W M W s W t (t) x x c, x f x Ɣ x 1, x 2 x y z = x + iy z = x iy z i Virtual work of cohesive force Virtual work of external loading Gauss weight factor Radial weight function at a Gauss point g Internal virtual work Interaction work Strain energy Time weight function Position vector Position vector for coarse and fine scales Position of projection point on an interface Two-dimensional coordinate system Local crack tip coordinate axes Local crack tip coordinate axes Complex variable Conjugate complex variable Complex parameters f, f First and second derivative of a function f, f First and second integrals of a function = / x Nabla operator Jump operator across an interface BEM CBS COD CTOD DCT DOF EDI EFG ELM EPFM FDM FE FEM FMM FPM FPZ GFEM HRR LEFM LSM MCC MEPU MLPG MLS Boundary Element Method Characteristic Based Split Crack Opening Displacement Crack Tip Opening Displacement Displacement Correlation Technique Degree Of Freedom Equivalent Domain Integral Element-Free Galerkin Equilibrium On Line Elastic Plastic Fracture Mechanics Finite Difference Method Finite Element Finite Element Method Fast Marching Method Finite Point Method Fracture Process Zone Generalised Finite Element Method Hutchinson Rice Rosengren Linear Elastic Fracture Mechanics Level Set Method Modified Crack Closure Multiscale Enrichment Partition of Unity Meshless Local Petrov Galerkin Moving Least Squares

17 Nomenclature xvii NURBS OUM PU PUFEM RKPM SAR SIF SPH TFEM TXFEM WLS XFEM, X-FEM Non-Uniform Rational B-Spline Ordered Upwind Method Partition of Unity Partition of Unity Finite Element Method Reproducing Kernel Particle Method Statically Admissible stress Recovery Stress Intensity Factor Smoothed Particle Hydrodynamics Time Finite Element Method Time extended Finite Element Method Weighted Least Squares extended Finite Element Method