Table Of ContentLecture Notes in Applied Mechanics
Volume 7
Series Editor
Prof. Dr.-Ing. Friedrich Pfeiffer
Springer
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Elastic-Plastic
Mixed-Mode Fracture
Criteria and Parameters
Valery N. Shlyannikov
Springer
Professor
VALERY N. SHLYANNIKOV
Kazan State Power Engineering University
Krasnoselskaya Street 51
420066 Kazan
Russia
e-mail: [email protected]
ISBN 978-3-642-53659-5
Library of Congress Cataloging-in-Publication Data
Shlyannikov, Valery N., 1953 -
Elastic-plastic mixed-mode fracture criteria and parameters / Valery N. Shlyannikov.
p. cm. - (Lecture notes in applied mechanics; v. 7)
Includes bibliographical references and index.
ISBN 978-3-642-53659-5 ISBN 978-3-540-45836-4 (eBook)
DOI 10.1007/978-3-540-45836-4
1. Fracture mechanics. 2. Elastoplasticity. I. Title. II. Series.
TA409 .S53 2002
620.1' 126-dc21 2002030950
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CONTENTS
I. Mixed-mode crack behavior under plane stress and plane strain small scale
yielding ....................................................................................................................... 1
1.1 Governing equations ........................................................................................ 3
1.1.1 Plane strain ............................................................................................... 4
1.1.2 Planc stress ............................................................................................... 7
1.1.3 Boundary conditions for different types of dominating fracture
mechanism ......................................................................................................... 9
1.2 Numerical iterative method for solving the nonlinear eigenvalue
problems ......................................................................................................... 10
1.3 Application of l-integral to plastic stress intensity factor determination ....... 14
1.3.1 Plane strain ............................................................................................. 15
1.3.2 Plane stress ...................................................................... 16
104 Family of crack-tip fields characterized by dominating fracture
mechanism ..................................................................................................... 18
104.1 Plane strain ............................................................................................. 18
104.2 Plane stress ........................................................................ 35
1.5 Finite element analysis of stress distributions at the crack tip .................... 53
1.6 Conditions of existence for mixed mode fracture ................................. 61
II. Modeling of crack growth by fracture damage zone ...................................... 73
2.1 A modified strain-energy density approach ................................................... 75
2.l.l Elastic strain energy density ................................................................... 76
2.1.2 Plastic strain energy density ................................................................... 78
2.2 Strain energy density distributions ................................................................. 80
2.3 Fracture damage zone .................................................................................... 90
2.3.1 A briefreview ......................................................................................... 90
2.3.2 Fracture damage zone size .................................................................... 102
204 Relation between cracks growth resistance and fracture process parameters
in elastic-plastic solids ............................................................... 106
2.5 Elastic-plastic approach for modeling of fatigue crack behavior ............... 116
2.6 Some aspects of the fatigue crack path prediction ............................... 126
III. Experimental investigation of fatigue crack propagation ........................... 131
3.1 Specimens for study of fatigue and fracture processes and material
properties ............................................................................. 132
3.2 Method of interpretation for cyclic crack resistance characteristics .......... 142
3.3 Effect of biaxial stress on fatigue crack growth in aluminum alloys ......... 150
3.4 Influence of mixed mode loading on fatigue fracture of high strength
steels .................................................................................. 159
3.5 Fatigue crack growth trajectories for the aluminum alloys and steels ........ 163
IV. Models for predicting crack growth rate and fatigue life ............................ 171
4.1 Crack growth direction criterion .................................................................. 172
4.2 Criteria of equivalent plastic strain under a complex stress state ................. 180
4.3 A model for predicting crack growth rate under biaxial loads ................ 189
4.4 An analysis of crack growth under complex stress state with taking into
account their orientation ............................................................ 195
V. Practical applications ....................................................................................... 203
5.1 Fracture analysis of gas turbine engine disks and simulation modeling of
operational conditions .................................................................................. 204
5.1.1 Stress state analysis ............................................................ 205
5.1.2 Crack growth model ............................................................................. 213
5.1.3 Full size disk experiments .................................................................... 215
5.2 Modeling fatigue crack behavior in a pressurized cylinder. .................. 220
5.2.1 Crack growth model ............................................................................. 222
5.2.2 Results and discussion ......................................................... 225
Reference ................................................................................................................ 235
Index ....................................................................................................................... 244
Foreword
My wife Tatyana,
daughter Mariya,
son Alexandr
It is well known that the mixed-mode conditions appear when the direction of the
applied loading does not coincide with the orthogonal K,-Kn-Km space. In general,
in the industrial practice the mixed-mode fracture and the mixed-mode crack
growth are more likely to be considered the rule than the exception. Miller et al.
considers that cracks can grow due to a mixture of processes (ductile and brittle),
mechanisms (static, fatigue, creep) and loading modes (tension, torsion, biax
ial/multiaxial). Additionally mixed-mode crack-extension can be affected by many
other considerations such as artifact geometry (thin plates, thick shells, and the
size, shape and orientation of the defect), environmental effects (temperature,
gaseous and liquid surroundings), material state (crystallographic structure, heat
treatment and route of manufacture) and stress conditions (out-of-phase and ran
dom loading effects).
The main feature of the mixed-mode fracture is that the crack growth would no
longer take place in a self-similar manner and does not follow a universal trajec
tory that is it will grow on a curvilinear path. There are various fracture criteria,
which predict the behavior of cracks in brittle and ductile materials loaded in
combined modes. Linear elastic fracture mechanics (LEFM) criteria predict basi
cally the same direction for crack propagation. Cracks in brittle materials have
been shown to propagate normal to the maximum tangential stress. In ductile ma
terials yielding occurs at the crack tip and LEFM is no longer applicable. The ap
plication of linear elastic fracture mechanics or elastic-plastic fracture mechanics
depends on the plastic zone size with respect to the crack length.
Despite of numerous elastic-plastic finite element analyses, experimental inves
tigations and have developed appropriate experimental techniques, our under
standing of both physics and mechanics of mixed-mode fracture phenomena is far
from complete. The main idea of this book is the realization of a common ap
proach that by means of corresponding models sets an interrelation between the
processes occurring on both micro- and macroscopic scale level with respect to
material structure. This approach allows the elastic-plastic analysis of accumulates
damage and fracture to make practical application of engineering materials and
structures fatigue life estimations.
The generalization of the HRR-approach in small scale yielding conditions that
has already become classic is rendered in the first chapter. A finite element inves
tigation of the effect of crack tip constraint under mixed mode large scale yielding
represents the issue of the day, which we intend to give in the following substan
tive work. Our investigation has given rise to a new field: analytical study of tran
sition of dominant fracture mechanism, in which tensile and shear cracks are con
sidered. The emphasis of this study is on the new approaches to the mixed-mode
problem in fatigue and fracture, and in particular the fracture damage zone ap
proach. The experimental investigations in the given book playa significant role,
therefore results of elastic-plastic crack growth under mixed mode I and II for six
types of the aluminum alloys and three types of steel were presented. The results
of experimental and theoretical researches are the background for crack growth
prediction models. Two models for predicting the crack growth rate and fatigue
life of an initially angled crack under biaxial loads of arbitrary direction are sug
gested. Special attention in this book is paid to practical application of the sug
gested models. Fatigue crack growth under operation conditions for rotating disks
of aircraft gas turbine engines is analyzed. Another part of the practical applica
tions is devoted to pressure vessels and piping.
The subject matter and the contents of the book reflect the scientific interests of
the author and the experience gained during the collaboration with the enterprises
of the aerospace industry. The course of lectures in Kazan State Power Engineer
ing University on Elementary theory of elasticity, plasticity and creep, Computa
tional solids and fracture mechanics, Modeling fatigue and fracture engineering
materials and structures is given by the author with the use of this material. And
finally, the author would like to express his gratitude to his colleagues, who sup
ported him with the carrying out of the sections of the conjoint works.
Valery Shlyannikov
I. Mixed-mode crack behavior under plane stress
and plane strain small scale yielding
One of the important points is that, for a large number of mixed-mode crack
growth problems of which we are aware, there are two fundamentally distinct
classes of growth: maximum principal stress-dominated and shear-dominated.
This is true regardless of whether we consider static or cyclic loading conditions.
Another point is the intimate connection of the crack tip displacement concept to
mixed-mode elastic-plastic fracture and fatigue processes. Several elastic-plastic
finite element analyses (Aoki et al. [7], Budden [23,24], Dhirendra and Narasim
han [35], Arun Roy and Narasimhan [125], Xiaosheng Gao and Shih [53]) and ex
perimental investigations (Bhattacharjee and Knott [12], Daile Donne and Doker
[34], Maccagno and Knott [91], Shlyannikov [135,136], Swankie and Smith
[156]) showed non-uniform deformation and damage fields near an initially
smooth notch tip under mixed mode loading.
Aoki et al. [7] predicted that two competing process zones might be associated
with the crack tip; one process zone, dominated by tensile stress and the other
dominated by shear stress. The side of the notch, dominated by tensile stress,
blunts, while the other side, dominated by shear strains, sharpens. It is possible
that material failure due to shear crack propagation in the direction of maximum
shear strains would occur in the localized band of intense plastic strain i.e. copla
nar with the notch plane (referred to in (DaIle Donne and Doker [34]) as "shear
crack"). This obviously will aid shear strain to concentrate at the sharpened side
and crack initiation. The highest tensile hydrostatic stress and notch-tip constraint
always occur near the blunted part of the notch. In material with higher work
hardening capacity the strains will be more widely distributed and hence the prob
ability of void nucleation to occur at any point near the notch tip will be almost
equal. Since crack extension finally occurs by growth of the voids and their coa
lescence, those voids subjected to the maximum tensile hydrostatic stresses will
grow coalesce first. Since the tensile hydrostatic stress is maximum near the
blunted side crack, initiation is likely to occur at that site. In this region the crack
growth direction is normal to the maximum tensile stresses. This type of mixed
mode ductile fracture mechanism is referred to in (DaIle Donne and Doker [34])
as "tensile crack" growth.
Thus, there are two competing fracture mechanisms that are operative near the
sharpened and blunted site of the notch respectively in a ductile material under
mixed-mode loading. Moreover, the mixed mode ratio will certainly have an ef
fect and transition at the site of crack initiation may be observed with change of
loading conditions. The dominant mechanism (of the two) determines the stable
