EXTENDED FINITE ELEMENT METHOD J+J,W5t^
for Fracture Analysis of Structures %)x9u$4W2
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by +t+<?M B
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Soheil Mohammadi $+!dP{
School of Civil Engineering pW (rNAJ!
University of Tehran Ve3z5d:^
Tehran, Iran |4Q *4s
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Published by Blackwell Publishing Ltd 2008 Aw |;C
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Progressive failure/fracture analysis of structures has been an active research topic for |)29"_Kk5
the past two decades. Historically, it has been addressed either within the framework pn
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of continuum computational plasticity and damage mechanics, or the discontinuous .h7s.p?
approach of fracture mechanics. The present form of linear elastic fracture mechanics $/++afi m
(LEFM), with its roots a century old has since been successfully applied to various ~ELMLwn.
classical crack and defect problems. Nevertheless, it remains relatively limited to simple '7-Yo
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geometries and loading conditions, unless coupled with a powerful numerical tool such :oP LluW*
as the finite element method and meshless approaches. hM Dd*<%l
The finite element method (FEM) has undoubtedly become the most popular and U]iI8c
powerful analytical tool for studying a wide range of engineering and physical problems. h'):/}JPl
Several general purpose finite element codes are now available and concepts of ["Ltqgx
FEM are usually offered by all engineering departments in the form of postgraduate d,V#5l-6
and even undergraduate courses. Singular elements, adaptive finite element procedures, sU ZA!sv
and combined finite/discrete element methodologies have substantially contributed to N1c=cZDV
the development and accuracy of fracture analysis of structures. Despite all achievements, PgWWa*Ew
the continuum basis of FEM remained a source of relative disadvantage for ;u UFgDi
discontinuous fracture mechanics. After a few decades, a major breakthrough seems YAr6 cl
to have been made by the fundamental idea of partition of unity and in the form of the elOeXYO0
eXtended Finite Element Method (XFEM). 9=%zd z2_S
This book has been prepared primarily to introduce the concepts of the newly N{;!xI v
developed extended finite element method for fracture analysis of structures. An attempt {LO Pm1K8Y
has also been made to discuss the essential features of XFEM for other related Cbw *? 9d
engineering applications. The book can be divided into four parts. The first part is dedicated E-bswUVaEE
to the basic concepts and fundamental formulations of fracture mechanics. It hmO2s/~
covers discussions on classical problems of LEFM and their extension to elastoplastic )/H;5 cn
fracture mechanics (EPFM). Issues related to the standard finite element modelling n/+X3JJ
of fracture mechanics and the basics of popular singular finite elements are reviewed g* q# VmE
briefly. ,,'jyqD
The second part, which constitutes most of the book, is devoted to a detailed discussion 5 p;AO N
on various aspects of XFEM. It begins by discussing fundamentals of partition 94u{k 1d x
of unity and basics of XFEM formulation in Chapter 3. Effects of various enrichment t'eqk#rq
functions, such as crack tip, Heaviside andweak discontinuity enrichment functions are VY0.]t
also investigated. Two commonly used level set and fast marching methods for tracking ]pax,| +$C
moving boundaries are explained before the chapter is concluded by examining a Z*%;;&?
number of classical problems of fracture mechanics. The next chapter deals with the k~ E x_2;#
orthotropic fracture mechanics as an extension of XFEM for ever growing applications m[9.'@ ye
of composite materials. A different set of enrichment functions for orthotropic media v= 5 5{
is presented, followed by a number of simulations of benchmark orthotropic problems. {3~VLdy
Chapter 5, devoted to simulation of cohesive cracks by XFEM, provides theoretical ^\\3bW9}H
bases for cohesive crack models in fracture mechanics, classical FEM and XFEM. _R4}\3}!
The snap-back response and the concept of critical crack path are studied by solving a Mt[yY|Ec|
number of classical cohesive crack problems. ToXWFX
The third part of the book (Chapter 6) provides basic information on new frontiers F "@% 7xy
of application of XFEM. It begins with discussions on interface cracking,which include I{Zb/}k-
classical solutions from fracture mechanics and XFEM approximation. Application of e~o!Qm
XFEM for solving contact problems is explained and numerical issues are addressed. N9e'jM>Oos
The important subject of dynamic fracture is then discussed by introducing classical b\k]Jx
formulations of fracture mechanics and the recently developed idea of time–space D* XrK0#Z`
discretization by XFEM. New extensions of XFEM for very complex applications of YG "Ta|@5
multiscale and multiphase problems are explained briefly. 2"Os9 KD
The final chapter explains a number of simple instructions, step-by-step procedures f-ltV<C_
and algorithms for implementing an efficient XFEM. These simple guidelines, in b$ G {^
combination with freely available XFEM source codes, can be used to further advance t un}rdb
the existing XFEM capabilities. bjZJP\6
This book is the result of an infinite number of brilliant research works in the ~wvt:E,f C
field of computational mechanics for many years all over the world. I have tried to "[ bkdL<
appropriately acknowledge the achievements of corresponding authors within the text, zA,vp^
relevant figures, tables and formulae. I am much indebted to their outstanding research 7FB?t<x
works and any unintentional shortcoming in sufficiently acknowledging them is sincerely I= mz^c{
regretted. Perhaps such a title should have become available earlier by one of &OMlW _FHR
the pioneers of the method, i.e. Professor T. Belytschko, a shining star in the universe mxa ~JAlN_
of computational mechanics, Dr J. Dolbow, Dr N. Mo¨es, Dr N. Sukumar and possibly j#NyNv(jE1
others who introduced, contributed and developed most of the techniques. p~pD`'%
I would like to extend my acknowledgement to Blackwell Publishing Limited, j{@O %fv=
for facilitating the publication of the first book on XFEM; in particular N. Warnock- P*|N)S)X%
Smith, J. Burden, L. Alexander, A. Cohen and A. Hallam for helping me throughout %go2tv:|W
the work. Also, I would like to express my sincere gratitude to my long-time friend, t*{L[c9.Uq
Professor A.R. Khoei, with whom I have had many discussions on various subjects of P}bIp+
computational mechanics, including XFEM. Alsomy special thanks go tomy students: %b6$N_M{H1
Mr A. Asadpoure, to whom I owe most of Chapter 4, Mr S.H. Ebrahimi for solving \Z-th,t
isotropic examples in Chapter 3 and Mr A. Forghani for providing some of the results E
C?}iP
in Chapter 5. >p_W(u@ z$
This book has been completed on the eve of the new Persian year; a ‘temporal twT/uBQ4a
interface’ between winter and spring, and an indication of the beginning of a blooming <_E KCk
season for XFEM, I hope. Z%t_1t
Finally, I would like to express my gratitude to my family for their love, understanding a)_ 3r]sv^
and never-ending support. I have spent many hours on writing this book; hours })g<I+]Hf9
that could have been devoted to my wife and little Sogol: the spring flowers that inspire ^7gG tz2
the life. &?<uR)tl
Soheil Mohammadi @Jt$92i5PS
Tehran, Iran (jc@8 @Wo.
Spring 2007
