The Variational Approach to Fracture
One of the goals of the Journal of Elasticity: The Physical and Ma- ematical Science of Solids is to identify and to bring to the attention of the research community in the physical and mathematical sciences extensive expositions which contain creative id
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THE VARIATIONAL APPROACH TO FRACTURE
Blaise BOURDIN Louisiana State University
Gilles A. FRANCFORT Université Paris-Nord
Jean-Jacques MARIGO Université Pierre et Marie Curie
Blaise Bourdin Louisiana State University Gilles A. Francfort Université Paris-Nord Jean-Jacques Marigo Université Pierre et Marie Curie
Library of Congress Control Number: 2008921762 ISBN 978-1-4020-6394-7
e-ISBN 978-1-4020-6395-4
© 2008 Springer Science+Business Media B.V. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper. 9 8 7 6 5 4 3 2 1 springer.com
The variational approach to fracture
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Foreword
One of the goals of the Journal of Elasticity: The Physical and Mathematical Science of Solids is to identify and to bring to the attention of the research community in the physical and mathematical sciences extensive expositions which contain creative ideas, new approaches and current developments in modelling the behaviour of materials. Fracture has enjoyed a long and fruitful evolution in engineering, but only in recent years has this area been considered seriously by the mathematical science community. In particular, while the age-old Griffith criterion is inherently energy based, treating fracture strictly from the point of view of variational calculus using ideas of minimization and accounting for the singular nature of the fracture fields and the various ways that fracture can initiate, is relatively new and fresh. The variational theory of fracture is now in its formative stages of development and is far from complete, but several fundamental and important advances have been made. The energy-based approach described herein establishes a consistent groundwork setting in both theory and computation. While it is physically based, the development is mathematical in nature and it carefully exposes the special considerations that logically arise regarding the very definition of a crack and the assignment of energy to its existence. The fundamental idea of brittle fracture due to Griffith plays a major role in this development, as does the additional dissipative feature of cohesiveness at crack surfaces, as introduced by Barenblatt. The following invited, expository article by B. Bourdin, G. Francfort and J.-J. Marigo represents a masterful and extensive glimpse into the fundamental variational structure of fracture. It contains examples from both theory and computation, and it suggests a related, introductory extension into the field of fatigue.
Minneapolis, 2007
Roger Fosdick
The variational approach to fracture
vii
Preface
In this tract, we wish to offer a panorama of the variational approach to brittle fracture that has developed in the past eight y
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