EXTENDED FINITE ELEMENT METHOD / 6+NU^
for Fracture Analysis of Structures |Au ]1 }
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Soheil Mohammadi x\\~SGd
School of Civil Engineering PRB{VC<k
University of Tehran l= }~v
Tehran, Iran 6l\FIah@
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Published by Blackwell Publishing Ltd 2008 =ll=)"O
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Progressive failure/fracture analysis of structures has been an active research topic for v [njdP
the past two decades. Historically, it has been addressed either within the framework k2 Q
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of continuum computational plasticity and damage mechanics, or the discontinuous 89~ =e Y
approach of fracture mechanics. The present form of linear elastic fracture mechanics d s:->+ o
(LEFM), with its roots a century old has since been successfully applied to various # 1.YKo
classical crack and defect problems. Nevertheless, it remains relatively limited to simple TDt Amk
geometries and loading conditions, unless coupled with a powerful numerical tool such 'h1b1,b~
as the finite element method and meshless approaches. 1HT_
The finite element method (FEM) has undoubtedly become the most popular and o0It82?RN
powerful analytical tool for studying a wide range of engineering and physical problems. k`{7}zxS
Several general purpose finite element codes are now available and concepts of k\`S
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FEM are usually offered by all engineering departments in the form of postgraduate NbRn*nb/T
and even undergraduate courses. Singular elements, adaptive finite element procedures, I?Aj.{{$G%
and combined finite/discrete element methodologies have substantially contributed to b)# Oc,
the development and accuracy of fracture analysis of structures. Despite all achievements, n_ nl{
the continuum basis of FEM remained a source of relative disadvantage for iK()&TNz
discontinuous fracture mechanics. After a few decades, a major breakthrough seems V5`^Y=X(%
to have been made by the fundamental idea of partition of unity and in the form of the [I;^^#'P
eXtended Finite Element Method (XFEM). MXD4|r(
This book has been prepared primarily to introduce the concepts of the newly !j:`7PT\
developed extended finite element method for fracture analysis of structures. An attempt "xJ 0 vlw
has also been made to discuss the essential features of XFEM for other related eH%RNtP`
engineering applications. The book can be divided into four parts. The first part is dedicated qe$^q
to the basic concepts and fundamental formulations of fracture mechanics. It w
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covers discussions on classical problems of LEFM and their extension to elastoplastic PVOx`<ng
fracture mechanics (EPFM). Issues related to the standard finite element modelling Dw<k3zaW
of fracture mechanics and the basics of popular singular finite elements are reviewed x_#-tB
briefly. @>Mxwpl?
The second part, which constitutes most of the book, is devoted to a detailed discussion O) atNE
on various aspects of XFEM. It begins by discussing fundamentals of partition tBATZ0nK`Q
of unity and basics of XFEM formulation in Chapter 3. Effects of various enrichment .dMVoG5
functions, such as crack tip, Heaviside andweak discontinuity enrichment functions are *^%*o?M~
also investigated. Two commonly used level set and fast marching methods for tracking _-TA{21)
moving boundaries are explained before the chapter is concluded by examining a >rsqH+oL
number of classical problems of fracture mechanics. The next chapter deals with the 3< Od0J
orthotropic fracture mechanics as an extension of XFEM for ever growing applications e8= YGx^o`
of composite materials. A different set of enrichment functions for orthotropic media PU<PhuMd
is presented, followed by a number of simulations of benchmark orthotropic problems. gB'fFkd
Chapter 5, devoted to simulation of cohesive cracks by XFEM, provides theoretical 9W ^xlid6
bases for cohesive crack models in fracture mechanics, classical FEM and XFEM. a2 +~;{?g
The snap-back response and the concept of critical crack path are studied by solving a {a(&J6$VE
number of classical cohesive crack problems. "Z=5gj
The third part of the book (Chapter 6) provides basic information on new frontiers [K / m
of application of XFEM. It begins with discussions on interface cracking,which include >Z.\J2wM<j
classical solutions from fracture mechanics and XFEM approximation. Application of rQ`i8GF
XFEM for solving contact problems is explained and numerical issues are addressed. =
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The important subject of dynamic fracture is then discussed by introducing classical )!BsF'uVQ
formulations of fracture mechanics and the recently developed idea of time–space }J:+{4Yn
discretization by XFEM. New extensions of XFEM for very complex applications of ~]C%/gEh
multiscale and multiphase problems are explained briefly. Hu3wdq
The final chapter explains a number of simple instructions, step-by-step procedures )\uO9PB[O
and algorithms for implementing an efficient XFEM. These simple guidelines, in *J|(jdu7
combination with freely available XFEM source codes, can be used to further advance \55VqGyxu9
the existing XFEM capabilities. NB_ )ZEmF
This book is the result of an infinite number of brilliant research works in the IuF_M<d,
field of computational mechanics for many years all over the world. I have tried to /4\wn?f
appropriately acknowledge the achievements of corresponding authors within the text, .
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relevant figures, tables and formulae. I am much indebted to their outstanding research kX`[Y@nUN
works and any unintentional shortcoming in sufficiently acknowledging them is sincerely -_ C#wtC
regretted. Perhaps such a title should have become available earlier by one of ) 0 W`
the pioneers of the method, i.e. Professor T. Belytschko, a shining star in the universe ikD1N
of computational mechanics, Dr J. Dolbow, Dr N. Mo¨es, Dr N. Sukumar and possibly Z6p5* +
others who introduced, contributed and developed most of the techniques. 1 rs&74-
I would like to extend my acknowledgement to Blackwell Publishing Limited, :#~U<C@o
for facilitating the publication of the first book on XFEM; in particular N. Warnock- \b=Pj!^gwb
Smith, J. Burden, L. Alexander, A. Cohen and A. Hallam for helping me throughout F[
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the work. Also, I would like to express my sincere gratitude to my long-time friend, $Fkaa<9;P
Professor A.R. Khoei, with whom I have had many discussions on various subjects of ]i<[d ,
computational mechanics, including XFEM. Alsomy special thanks go tomy students: C'yppl%
Mr A. Asadpoure, to whom I owe most of Chapter 4, Mr S.H. Ebrahimi for solving Cqi i}
isotropic examples in Chapter 3 and Mr A. Forghani for providing some of the results ?S$i?\ Qh
in Chapter 5. NEt1[2 X%
This book has been completed on the eve of the new Persian year; a ‘temporal ew _-Eb
interface’ between winter and spring, and an indication of the beginning of a blooming YKmsQ(q`N
season for XFEM, I hope. U6E\AvbRn
Finally, I would like to express my gratitude to my family for their love, understanding /H7&AiA
and never-ending support. I have spent many hours on writing this book; hours 5irwz4.4
that could have been devoted to my wife and little Sogol: the spring flowers that inspire 0& >H^
the life. l5xCz=dw
Soheil Mohammadi (Rt7%{*
Tehran, Iran ",aT WQgN
Spring 2007
