EXTENDED FINITE ELEMENT METHOD )O%lh
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for Fracture Analysis of Structures Dy^4^ J5+
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by 2=N YBOE
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Soheil Mohammadi
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School of Civil Engineering am05>c9
University of Tehran i&FC-{|Z
Tehran, Iran w G Q{
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Published by Blackwell Publishing Ltd 2008 @w:sNXz -
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Progressive failure/fracture analysis of structures has been an active research topic for Xl*-A|:j
the past two decades. Historically, it has been addressed either within the framework |q Nrj~n@
of continuum computational plasticity and damage mechanics, or the discontinuous n*6',BY
approach of fracture mechanics. The present form of linear elastic fracture mechanics V2]S{!p}k
(LEFM), with its roots a century old has since been successfully applied to various "WY cw\@U
classical crack and defect problems. Nevertheless, it remains relatively limited to simple +CNRSq"
geometries and loading conditions, unless coupled with a powerful numerical tool such U>x2'B v
as the finite element method and meshless approaches. C5RDP~au
The finite element method (FEM) has undoubtedly become the most popular and LDv F)Eg
powerful analytical tool for studying a wide range of engineering and physical problems. TJ5{Ee GV
Several general purpose finite element codes are now available and concepts of emS +%6U
FEM are usually offered by all engineering departments in the form of postgraduate y$V{yh[:
and even undergraduate courses. Singular elements, adaptive finite element procedures, p N]H p"v
and combined finite/discrete element methodologies have substantially contributed to )x| BY>
the development and accuracy of fracture analysis of structures. Despite all achievements, qc'tK6=jp
the continuum basis of FEM remained a source of relative disadvantage for 0I?3@Nz6
discontinuous fracture mechanics. After a few decades, a major breakthrough seems rb \Ohv\
to have been made by the fundamental idea of partition of unity and in the form of the ?3z+|;t6C
eXtended Finite Element Method (XFEM). IL:"]`f*
This book has been prepared primarily to introduce the concepts of the newly 5\Y/s o=
developed extended finite element method for fracture analysis of structures. An attempt -{b1&
has also been made to discuss the essential features of XFEM for other related 6eK^T=
engineering applications. The book can be divided into four parts. The first part is dedicated JkxS1
to the basic concepts and fundamental formulations of fracture mechanics. It '\*Rw]bR|
covers discussions on classical problems of LEFM and their extension to elastoplastic c[y=K)<Z
fracture mechanics (EPFM). Issues related to the standard finite element modelling pmW=l/6+V3
of fracture mechanics and the basics of popular singular finite elements are reviewed Ft.BfgJ$
briefly. j*:pW;)^
The second part, which constitutes most of the book, is devoted to a detailed discussion ?s"v0cg+
on various aspects of XFEM. It begins by discussing fundamentals of partition Z#MO Df0H@
of unity and basics of XFEM formulation in Chapter 3. Effects of various enrichment BtHvfoT
functions, such as crack tip, Heaviside andweak discontinuity enrichment functions are F<(x z=
also investigated. Two commonly used level set and fast marching methods for tracking AYZds >#Q
moving boundaries are explained before the chapter is concluded by examining a -6tF
number of classical problems of fracture mechanics. The next chapter deals with the rw\4KI@ L
orthotropic fracture mechanics as an extension of XFEM for ever growing applications >;3c; nf
of composite materials. A different set of enrichment functions for orthotropic media >6DY3\
is presented, followed by a number of simulations of benchmark orthotropic problems. hy )RV=X
Chapter 5, devoted to simulation of cohesive cracks by XFEM, provides theoretical nG%j4r ;
bases for cohesive crack models in fracture mechanics, classical FEM and XFEM. VD#^Xy4% r
The snap-back response and the concept of critical crack path are studied by solving a 8rpN2M 3h
number of classical cohesive crack problems. M y`%gP~%g
The third part of the book (Chapter 6) provides basic information on new frontiers
610k#$
of application of XFEM. It begins with discussions on interface cracking,which include cT0g, ^&
classical solutions from fracture mechanics and XFEM approximation. Application of }t-r:R$,
XFEM for solving contact problems is explained and numerical issues are addressed. M7> \Qk
The important subject of dynamic fracture is then discussed by introducing classical [ sk" 2
formulations of fracture mechanics and the recently developed idea of time–space eXaDx%m M
discretization by XFEM. New extensions of XFEM for very complex applications of `A^} X
multiscale and multiphase problems are explained briefly. gq@."wHU
The final chapter explains a number of simple instructions, step-by-step procedures .yP
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and algorithms for implementing an efficient XFEM. These simple guidelines, in gwYd4
combination with freely available XFEM source codes, can be used to further advance ^ KjqS\<
the existing XFEM capabilities. [1UqMkXtf
This book is the result of an infinite number of brilliant research works in the Nb9pdkf0
field of computational mechanics for many years all over the world. I have tried to )w` Nkx
appropriately acknowledge the achievements of corresponding authors within the text, Hf-F-~ E
relevant figures, tables and formulae. I am much indebted to their outstanding research (_08? cN
works and any unintentional shortcoming in sufficiently acknowledging them is sincerely jw[`_
regretted. Perhaps such a title should have become available earlier by one of 7=AKQ7BB>b
the pioneers of the method, i.e. Professor T. Belytschko, a shining star in the universe 5#F+-9r
of computational mechanics, Dr J. Dolbow, Dr N. Mo¨es, Dr N. Sukumar and possibly YaT07X.(b
others who introduced, contributed and developed most of the techniques. B&$89] gs|
I would like to extend my acknowledgement to Blackwell Publishing Limited, 5Q}@Y3 i=
for facilitating the publication of the first book on XFEM; in particular N. Warnock- si;]C~X*
Smith, J. Burden, L. Alexander, A. Cohen and A. Hallam for helping me throughout D JW1kR
the work. Also, I would like to express my sincere gratitude to my long-time friend, &L?Dog o
Professor A.R. Khoei, with whom I have had many discussions on various subjects of 7f$Lb,\y
computational mechanics, including XFEM. Alsomy special thanks go tomy students: \~H"!vj
Mr A. Asadpoure, to whom I owe most of Chapter 4, Mr S.H. Ebrahimi for solving :ZIcWIV-
isotropic examples in Chapter 3 and Mr A. Forghani for providing some of the results M:SxAo-D2
in Chapter 5. 09?<K)_G
This book has been completed on the eve of the new Persian year; a ‘temporal W[m _IY
interface’ between winter and spring, and an indication of the beginning of a blooming dCK -"#T!
season for XFEM, I hope. ]/'] {*T1
Finally, I would like to express my gratitude to my family for their love, understanding %% >?<4t
and never-ending support. I have spent many hours on writing this book; hours M vh_>-i
that could have been devoted to my wife and little Sogol: the spring flowers that inspire 3*TS
4xX
the life. } 00e @a
Soheil Mohammadi -&A[{m <,>
Tehran, Iran Mww]l[1'EL
Spring 2007
