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EXTENDED FINITE ELEMENT METHOD
for Fracture Analysis of Structures

by

Soheil Mohammadi
School of Civil Engineering
University of Tehran
Tehran, Iran

Published by Blackwell Publishing Ltd 2008

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 andweak 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
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. Mo¨es, 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. Alsomy special thanks go tomy 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

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