EXTENDED FINITE ELEMENT METHOD 3Z'{#<1>^;
for Fracture Analysis of Structures !)tXN=(1a
6<H[1PI`,G
by bIizh8d?
uMF\3T(x4
Soheil Mohammadi H<`[,t
School of Civil Engineering UQ>GAz h
University of Tehran L{2\NJ"+u
Tehran, Iran PK C``+K i
MAR
kTxzi
Published by Blackwell Publishing Ltd 2008 vnz.81OR
lg;Y}?P
Progressive failure/fracture analysis of structures has been an active research topic for #I@]8U#,":
the past two decades. Historically, it has been addressed either within the framework 5fb,-`m.
of continuum computational plasticity and damage mechanics, or the discontinuous HH6b{f@^
approach of fracture mechanics. The present form of linear elastic fracture mechanics p)ta c*US
(LEFM), with its roots a century old has since been successfully applied to various M@Q= !!tQ(
classical crack and defect problems. Nevertheless, it remains relatively limited to simple 9G_=)8sOV
geometries and loading conditions, unless coupled with a powerful numerical tool such 5S7`gN.
as the finite element method and meshless approaches. ^Gv<Xl
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. %e7{ke}r
Several general purpose finite element codes are now available and concepts of U=XaI%ZM)
FEM are usually offered by all engineering departments in the form of postgraduate )F? 57eh
and even undergraduate courses. Singular elements, adaptive finite element procedures, $E|W |4 N
and combined finite/discrete element methodologies have substantially contributed to |d D! @K
the development and accuracy of fracture analysis of structures. Despite all achievements, U^Z[6u
the continuum basis of FEM remained a source of relative disadvantage for QAi(uL5
discontinuous fracture mechanics. After a few decades, a major breakthrough seems N l_!%k:
to have been made by the fundamental idea of partition of unity and in the form of the vFb{(gIJ
eXtended Finite Element Method (XFEM). O)Nt"k7
b
This book has been prepared primarily to introduce the concepts of the newly ~N[hY1}X[
developed extended finite element method for fracture analysis of structures. An attempt '* >LZo4
has also been made to discuss the essential features of XFEM for other related ~B (]0:
engineering applications. The book can be divided into four parts. The first part is dedicated E yNI]XEj
to the basic concepts and fundamental formulations of fracture mechanics. It T9Vyj3!i_
covers discussions on classical problems of LEFM and their extension to elastoplastic :G^`LyO M
fracture mechanics (EPFM). Issues related to the standard finite element modelling g8XGZW!
of fracture mechanics and the basics of popular singular finite elements are reviewed }U'5j/EFZ
briefly. 6WfyP@ f
The second part, which constitutes most of the book, is devoted to a detailed discussion -f9M*7O<gf
on various aspects of XFEM. It begins by discussing fundamentals of partition 8tA.d .8
of unity and basics of XFEM formulation in Chapter 3. Effects of various enrichment c7s4 g-
functions, such as crack tip, Heaviside andweak discontinuity enrichment functions are ErESk"2t
also investigated. Two commonly used level set and fast marching methods for tracking ZX-9BJ`Q
moving boundaries are explained before the chapter is concluded by examining a 4Bx1L+Cg
number of classical problems of fracture mechanics. The next chapter deals with the d@At-Z~M
orthotropic fracture mechanics as an extension of XFEM for ever growing applications RMC|(Q<
of composite materials. A different set of enrichment functions for orthotropic media 6xL=JSi~
is presented, followed by a number of simulations of benchmark orthotropic problems. *,
*"G?
Chapter 5, devoted to simulation of cohesive cracks by XFEM, provides theoretical !u6~#.7
bases for cohesive crack models in fracture mechanics, classical FEM and XFEM. ~n[LL)v
The snap-back response and the concept of critical crack path are studied by solving a {]<D"x ;
number of classical cohesive crack problems. #]lUJ
&M}e
The third part of the book (Chapter 6) provides basic information on new frontiers ZX'{o9+w5
of application of XFEM. It begins with discussions on interface cracking,which include +8^9: w0}
classical solutions from fracture mechanics and XFEM approximation. Application of k|A! 5A2
XFEM for solving contact problems is explained and numerical issues are addressed. 'EZ[aY!);
The important subject of dynamic fracture is then discussed by introducing classical puqLXDjA/
formulations of fracture mechanics and the recently developed idea of time–space K+v 250J$-
discretization by XFEM. New extensions of XFEM for very complex applications of 2zFdKs,
multiscale and multiphase problems are explained briefly. u~s'<c+8_
The final chapter explains a number of simple instructions, step-by-step procedures Y sr{1! K
and algorithms for implementing an efficient XFEM. These simple guidelines, in B1 0p7+NBF
combination with freely available XFEM source codes, can be used to further advance 1
9
k$)m
the existing XFEM capabilities. T@yH. 4 D
This book is the result of an infinite number of brilliant research works in the a|nlmH"l
field of computational mechanics for many years all over the world. I have tried to sx]?^KR:
appropriately acknowledge the achievements of corresponding authors within the text, W$'pUhq\H
relevant figures, tables and formulae. I am much indebted to their outstanding research 9Biw!%a
works and any unintentional shortcoming in sufficiently acknowledging them is sincerely e<9nt [
regretted. Perhaps such a title should have become available earlier by one of w|L~+
the pioneers of the method, i.e. Professor T. Belytschko, a shining star in the universe #
Jdip)
of computational mechanics, Dr J. Dolbow, Dr N. Mo¨es, Dr N. Sukumar and possibly s=%HT fw
others who introduced, contributed and developed most of the techniques. Vj2GK"$v
I would like to extend my acknowledgement to Blackwell Publishing Limited, EW5S%Y
for facilitating the publication of the first book on XFEM; in particular N. Warnock- IN%> 46e`
Smith, J. Burden, L. Alexander, A. Cohen and A. Hallam for helping me throughout M%B[>pONb7
the work. Also, I would like to express my sincere gratitude to my long-time friend, _5`M( ;hL2
Professor A.R. Khoei, with whom I have had many discussions on various subjects of _23sIUN c3
computational mechanics, including XFEM. Alsomy special thanks go tomy students: E*w 2yWR
Mr A. Asadpoure, to whom I owe most of Chapter 4, Mr S.H. Ebrahimi for solving \^#1~Kx
isotropic examples in Chapter 3 and Mr A. Forghani for providing some of the results \Flq8S /t^
in Chapter 5. LpmspIPvf
This book has been completed on the eve of the new Persian year; a ‘temporal Ap!UX=HBb
interface’ between winter and spring, and an indication of the beginning of a blooming +XpRkX&-
season for XFEM, I hope. m Ksj7
Finally, I would like to express my gratitude to my family for their love, understanding `|/|ej]$P
and never-ending support. I have spent many hours on writing this book; hours "^XN"SUw
that could have been devoted to my wife and little Sogol: the spring flowers that inspire :.35pp,0
the life. ~HYP:6f
Soheil Mohammadi .oK7E(Q J
Tehran, Iran O]Kb~jkd
Spring 2007
