Modeling of Defects and Fracture Mechanics

211 Pages·1993·16.774 MB·English

by  G. Herrmann (eds.)

CISM COURSES AND LECTURES
Series Editors:
The Rectors of CISM
Sandor Kaliszky -Budapest
Horst Lippmann -Munich
Mahir Sayir- Zurich
The Secretary General of CISM
Giovanni Bianchi-Milan
Executive Editor
Carlo Tasso -Udine
The series presents lecture notes, monographs, edited works and proceedings in the field of
Mechanics, Engineering, Computer Science and Applied Mathematics.
Purpose of the series in to make known in the international scientific and technical
community results obtained in some of the activities organized by CISM, the International
Centre for Mechanical Sciences.
INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
COURSES AND LECfURES -No. 331
MODELING OF DEFECTS AND
FRACTURE MECHANICS
EDITED BY
G. HERRMANN
STANFORD UNIVERSITY, STANFORD
SPRINGER-VERLAG WIEN GMBH
Le spese di stampa di questo volume sono in parte coperte da
contributi del Consiglio Nazionale delle Ricerche.
This volume contains 75 illustrations.
This work is subject to copyright.
All rights are reserved,
whether the whole or part of the material is concerned
specifically those of translation, reprinting, re-use of illustrations,
broadcasting, reproduction by photocopying machine
or similar means, and storage in data banks.
© 1993 by Springer-Verlag Wien
Originally published by Springer Verlag Wien-New York in 1993
In order to make this volume available as economically and as
rapidly as possible the authors' typescripts have been
reproduced in their original forms. This method unfornmately
has its typographical limitations but it is hoped that they in no
way distract the reader.
ISBN 978-3-211-82487-0 ISBN 978-3-7091-2716-2 (eBook)
DOI 10.1007/978-3-7091-2716-2
PREFACE
In many industrial applications, materials and structures are subjected to various
manufacturing and service conditions which make it imperative to enhance the predictive
capabilities ofm odeling various types of defects. These include micro-cracks, micro-voids,
dislocations, etc., which precede possible gross fracture. And fracture processes
themselves, both static and dynamic, need to be modeled for various types of materials and
environments (e.g. elevated temperatures). At the intermediate stage between "small"
defects and flaws, sometimes preexisting, we find a range of processes commonly
described under the title of "damage mechanics".
To discuss the latest developments of mathematical modeling of these multifacetted
phenomena and processes, a Second International Summer School on Mechanics was held
at the International Center for Mechanical Sciences (CISM) in Udine, Italy, September 2-6,
I99I. It was sponsored by both ClSM and the International Union of Theoretical and
Applied Mechanics (IUTAM). The course was presented by 5 lecturers from 5 different
countries and attended by over 50 scientists and engineers from I7 different countries.
Thus it was a truly international week, much in the spirit of the two supporting
organizations.
It was of course realized that the set topic of "Modeling of Defect and Fracture
Mechanics" is of huge breath and width and could not be treated exhaustively by the five
speakers. Yet, we strived at maintaining a certain balance between the broad range of
phenomena observed experimentally, the mathematical modeling of physical events, and
the analytic and numerical treatment of their mathematical description.
To make sure that a common background was established at the very beginning, several
introductory lectures were delivered by Professor J. R. Willis, University of Bath, Great
Britain, which eventually, however, went well beyond an elementary review of continuum
mechanics by including elements ofb oth linear and nonlinear fracture mechanics.
The theme off racture mechanics was continued by Professor G. I. Barenblatt, USSR
Academy of Sciences (now G./. Taylor Professor in the University of Cambridge, Great
Britain), who emphasized similarity methods and renormalization groups applied to
fracture.
The theory of crystal defects and their impact on material behavior was articulated by
Professor E. Kroner, University of Stuttgart. Considered were theories of dislocations,
elastoplasticity as well as geometric theories.
Computational aspects of micromechanics were presented by Professor V.
Tvergaard, The Technical University of Denmark, Lyngby. In particular, numerical
treatment of ductile failure, plastic flow localization, creep rupture, debonding in metal
matrix composites and dynamic ductile crack growth were treated.
My own contribution focused on the "Application of the Heterogenization Methodology
to the Analysis of Elastic Bodies with Defects." This methodology is based on the
involution correspondence of elastic regions bounded by a circle and permits an efficient
analysis of elastic bodies with circular cavities or inclusions, in terms of correspondingly
loaded homogeneous bodies without such defects. Extensions from elasticity to
thermoelasticity and piezoelectricity were also included.
The local organization of the Summer School lay in the hands of ProfessorS. Kaliszky,
Rector ofCISM, and his staff, lead by Mrs. A. Bertozzi. We are grateful for all the help
and assistance they provided us. We also should acknowledge the patience of Professor C.
Tasso, editor of the CISM series of publications, in waiting an unduly long time for the
preparation of our manuscripts.
G. Herrmann
CONTENTS
Page
Preface
Introductory Lectures
by JR. Willis ......................................................................................... 1
Some General Aspects of Fracture Mechanics
by GJ. Barenblatt .................................................................................. 2 9
Theory of Crystal Defects and Their Impact on Material Behaviour
by E. Kroner ........................................................................................ 6 1
Computational Micromechanics
by V. Tvergaard .................................................................................... 1 1 9
Application of the Heterogenization Methodology to the Analysis
of Elastic Bodies with Defects
by G. Herrmann .................................................................................. 1 6 5
INTRODUTORY LECTURES
J.R. Willis
University of Bath, Bath, U.K.
2 J.R. Willis
X
0
0
Fig. 1.1 Sketch of a body in undeformed and deformed configurations.
1 Review of Deformation and Stress
This first section provides a reminder - for those that need it - of basic notions of
deformation and stress, relevant to all materials. A wide variety of notations is in
current use; here, the arbitrary choice has been made to follow, at least initially, that
used in the book by Ogden (1984).
Deformation
The deformation of a body is depicted in Fig. 1.1. It occupies a domain 8 in its
0
reference configuration and 81 currently; if the deformation varies with time t, then
81 depends on t. A generic point of the body has position vector X E 80 initially,
and x E 8 at time t, relative to origins 0 and o respectively. Relative to Cartesian
1
bases {Ea} for the initial configuration and { e;} currently, the vectors X and x have
coordinate representations
X = XaEa and X = x;e;, (1.1)
with i.mplied summation over the values 1,2,3 for the repeated suffixes.
The deformation is defined by an invertible map from 80 to 81• In terms of X and
x,
x = x(X,t) (1.2a)
Introductory Lectures 3
or, in components,
= Xi(X, t). ( 1.2b)
Xj
The deformation gradient A is then defined as
( 1.3)
where
( 1.4)
Then, an infinitesimal line segment dX deforms into the segment dx, where
dx = AdX, ( 1.5)
Strain tensors relate lengths and angles before and after deformation. If infinitesimal
line segments dX and dY transform respectively into dx and dy, then
dx.dy = dxT dy = dXT AT AdY. (1.6)
All information on length and angle changes is thus contained in AT A. Perhaps the
simplest strain measure - the Green strain - is then
(1.7)
A general class of strain measures is obtained by first defining the eigenvalues and
normalized eigenvectors of AT A as >.7 and u(i), so that
L3
AT A= >.7u(il 0 u(il. (1.8)
i=l
Then, iff is any monotone increasing function for which f(1) = 0 and f'(1) = 1, a
strain tensor e is defined as
L3
e = f(,\;)u(i) 0 u(i). (1.9)
i=l
The strain tensor (1.7) fits this pattern, with f(>.) = KV- 1).