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Proceedings of the European Mechanics Colloquium 239 'Mechanics of
Creep Brittle Materials' held at Leicester University, UK, 15-17
August 1988.
MECHANICS OF CREEP BRITTLE MATERIALS
1
A. C. F. COCKS and
A. R. S. PONTER Department oj Engineering, University oj Leicester,
UK
ELSEVIER APPLIED SCIENCE LONDON and NEW YORK
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WITH 25 TABLES AND 160 ILLUSTRATIONS
© 1989 ELSEVIER SCIENCE PUBLISHERS LTD © 1989 CENTRAL ELECTRICITY
GENERATING BOARD--pp. 13-35
© 1989 CROWN COPYRIGHT-pp. 99-Il6 © 1989 GOVERNMENT OF CANADA-pp.
201-212
Softcover reprint ofthe hardcover I st edition 1989
British Library Cataloguing in Publication Data Mechanics of creep
brittle materials I.
I. Materials. Creep I. Cocks, A.C.F. II. Ponter, A.R.S.
620.1'1233
ISBN-13: 978-94-010-6994-6 e-ISBN-13: 978-94-009-1117-8 001:
10.1007/978-94-009-1117-8
Library of Congress CIP data applied for
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damage to persons or property as a matter of products liability,
negligence or otherwise, or from any use or operation of any
methods,
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herein.
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v
Preface
Failure of components which operate in the creep range can result
either from the growth of a dominant crack or through the
accumulation of 'damage' in the material. Conventional and nuclear
power generating plant are generally designed on the basis of
continuum failure, with assessment routes providing an indication
of the effects of flaws on component performance. Another example
where an understanding of creep failure is important is in the
design of offshore structures which operate in arctic waters. These
structures can be subjected to quite considerable forces by
wind-driven ice sheets, which are limited by failure of the ice
sheet. Design codes are currently being developed which identify
the different mechanisms of failure, ranging from continuum
crushing to radial cracking and buckling of the ice sheet. Our
final example concerns engineering ceramics, which are currently
being considered for use in a wide range of high-temperature
applications. A major problem preventing an early adoption of these
materials is their brittle response at high stresses, although they
can behave in a ductile manner at lower stresses.
In each of the above situations an understanding of the processes
of fast fracture, creep crack growth and continuum failure is
required, and in particular an understanding of the material and
structural features that influence the transition from brittle to
ductile behaviour. The translation of this information to component
design is most advanced for metallic components. Research on ice
mechanics is largely driven by the needs of the oil industry, to
provide information on a limited class of problems. While, at the
present time, ceramic materials are still very much in the process
of development. Uncertainties in the reproducibility of physical
properties and the difficulties encountered in testing these
materials at elevated temperatures are hindering the development of
suitable design procedures.
The aim of Euromech Colloquium 239 was to bring together
researchers interested in the creep behaviour of metals,
engineering ceramics and ice to examine the processes of crack
growth and continuum failure. These proceedings are divided into
four sections, which examine either a particular type of failure
process, allowing comparisons to be made between the modelling of
different materials, or the behaviour of a particular class of
materials. Each section contains a selection of papers which
discuss the material phenomena, the
Vl
development of material models and the application of these models
to practical situations. The first section examines the processes
of crack propagation. This is followed by two sections devoted to
the behaviour of engineering ceramics and ice, with a final section
on continuum damage mechanics. This grouping of papers is by no
means exclusive and many of the papers which have been assigned to
one section could equally well have appeared in another.
It is evident from the papers presented in this volume and from the
lively discussions which accompanied each session of the Colloquium
that we can learn a great deal from the activities of researchers
working on related problems in different fields of study. We would
therefore encourage the reader not only to read the papers that
relate directly to his own research interests, but also to examine
the papers which, at first sight, might appear to be outside his
field of study.
We would like to take this opportunity to thank all those people
who helped to make the Colloquium a success. We are grateful to Sue
Ingle, Tim Wragg and their staff in the University Conference
Office and at Beaumont Hall for providing a welcoming, relaxed
environment and ensuring that the Colloquium ran smoothly. Our
thanks are also extended to Paul Smith for ensuring that none of
the presentations was disrupted by problems with audio-visual
equipment. We are particularly indebted to Jo Denning for all the
time and effort she put into the preparations for the Colloquium,
and for looking after the needs of the delegates, allowing us to
participate fully in the proceedings.
A. C. F. COCKS
A. R. S. PaNTER
University of Leicester, UK
1. Crack Propagation in Creeping Bodies
The brittle-to-ductile transition in silicon ..... . P. B. Hirsch,
S. C. Roberts,]. Samuels and P. D. Warren
Stress redistribution effects on creep crack growth R. A.
Ainsworth
Contour integrals for creep crack growth analysis W. S.
Blackburn
Modelling of creep crack growth C. A. Webster
M0delling creep-crack growth processes in ceramic materials M. D.
Thouless
On the growth of cracks by creep in the presence of residual
stresses D.]. Smith
2. Deformation and Failure of Engineering Ceramics
Creep deformation of engineering ceramics B. Wilshire
Statistical mapping and analysis of engineering ceramics data ]. D.
Snedden and C. D. Sinclair
Indentation creep in zirconia ceramics between 290 K and 1073 K ].
L. Henshall, C. M. Carter and R. M. Hooper
V
13
22
36
50
63
75
99
117
YI11
Ductile creep cracking in uranium dioxide T. E. Chung and T. j.
Davies
Physical interpretation of creep and strain recovery of a glass
ceramic near
129
glass transition temperature .. . . . . . . . . . . . . . . . . . .
141 C. Mai, H. Satha, S. Etienne andj. Pere;:;
3. Ice Mechanisms and Mechanics
Ice loading on offshore structures: the influence of ice strength
M. R. Mills and S. D. Hallam
Ice forces on wide structures: field measurements at Tarsuit Island
A. R. S. Ponter and P. R. Brown
The double torsion test applied to fine grained freshwater columnar
ice,
152
168
and sea ice. . . . . . . . . . . . . . . . . . . . . . . . . . 188
B. L. Parsons,j. B. Snellen and D. B. Muggeridge
Ice and steel: a comparison of creep and failure N. K. Sinha
. . . . . . . . . . . 201
A micromechanics based model for the creep of ice including the
effects of general microcracking . . . . . . . . . . . . . . . . .
. . . . . . 213
A. C. F. Cocks
Continuum damage mechanics applied to multi-axial cyclic material
behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . .
230
D. A. Lavender and D. R. Hayhurst
Multiaxial stress rupture criteria for ferritic steels . . . . . .
. . . . . 245 P. F. Aplin and G. F. Eggeler
Segregation of impurities in a heat-affected and an intercritical
zone in an operated O.SCr O.SMo 0.2SV steel . . . . . . . . . . . .
. . . . . . 262
P. Battaini, D. D'Angelo, A. Olchini and F. Parmigiani
Effect of creep cavitation at sliding grain boundaries E. van der
Giessen and V. Tvergaard
......... 277
THE BRITTLE-TO-DUCTILE TRANSITION IN SILICON
P.B. HIRSCH, S.G. ROBERTS, J. SAMUELS AND P.D. WARREN Department of
Metallurgy and Science of Materials
University of Oxford, Parks Road, Oxford OXl 3PH, UK
ABSTRACT
Recent experiments on the brittle-ductile transition (BDT) of
precracked specimens of Si show that the transition is sharp, and
that the strain rate dependence of the transition temperature, Te ,
is controlled by dislocation velocity. Etch pit observations show
that dislocation generation from the crack tip begins at K just
below Kre , from a small number of sources around the crack tip.
The dynamics of plastic relaxation has been simulated on a model in
which a small number of crack-tip sources operate and shield the
crack. The model predicts cleavage after some plasticity, and that
a sharp transition is obtained only if crack-tip sources are
nucleated at K=Ko just below Kre , and if these sources operate at
K=KN«Ko . A mechanism for the formation of crack-tip sources by the
movement of existing dislocations to and interaction with the crack
tip is proposed. The model predicts a dependence of Te and of the
shape of the BDT on the existing dislocation distribution, and this
has been confirmed by experiment.
1. INTRODUCTION
This paper presents results of recent experiments on the
brittle-to-ductile transition (BOT) in silicon. At the BDT plastic
relaxation processes blunt and shield the crack making crack
propagation more difficult, leading to an increase in fracture
stress with increasing temperature. The brittle-to ductile
transition temperature, Te , depends on strain rate, the activation
energy controlling Te being that for dislocation velocity. A
computer model simulating the dynamics of dislocation generation at
crack tips has been developed and the predictions of this model
have been compared with experiment.
2. EXPERIMENTAL APPROACH
Mechanical tests have been carried out using four-point bending of
precracked bar-shaped speCimens of float zone Si, with their long
axis (25mm) parallel to [111] and their shorter axes along [110]
(lmm) and [112] (3mm) respectively. The intended fracture plane,
perpendicular to the
2
direction of applied tensile stress, was a (Ill) plane, a natural
cleavage plane in Si. The sharp precrack was introduced by Knoop
indentation at room temperature. Crack depths of l3~m and 37~m were
used. This technique also leaves a plastic zone in the region of
the indentation; the residual stress was relaxed by annealing the
crystals at 800°C in vacuum.
3. EXPERIMENTAL RESULTS
Fig. 1 shows fracture stress against temperature for a given strain
rate, for intrinsic Si. The transition is extremely sharp. The
range of temperatures from the highest at which a specimen
fractures in a completely brittle manner to the lowest at which a
specimen deforms plastically is typically about 10°C.
~e :z
CD~ 4
b Brittle
o
Ductile
• • 6 0
Figure 1. Failure stress vs. temperature for intrinsic silicon
specimens tested at the minimum strain rate, 1.3xl0- 6S- 1 • Note
the sharpness of the
brittle-ductile transition.
The transition temperature Tc is strongly strain-rate dependent,
varying by about 100°C when the strain-rate is changed by a factor
10. Fig. 2 shows the results of tests carried out at different
strain rates, for intrinsic (2.5 x 1013 P atoms cm- 3 ) and n-type
material (2 x 1018 P atoms cm- 3 ). The precrack depth is l3~m in
all experiments except for point C, where the crack depth is 37~m.
The strain rate is expressed in terms of rate of increase of stress
intensity factor, K, using the expression of Newman and Raju [lJ
for a semicircular crack, and the relation between stress and
strain for a perfectly elastic beam in four-point bending.
Fig. 2 shows that K a exp-Ue/kTc ' where Ue is the experimental
activation energy. The values of the experimental activation energy
agree (within experimental error) with those determined by George
and Champier [2J for dislocation motion in similarly doped silicon
specimens. This confirms the original suggestion of St.John [3J
that the activation energy controlling the strain-rate dependence
of Tc is that for dislocation veloci ty. Fig. 2 also shows St.
John's original data for intrinsic Si, obtained using a tapered
double cantilever technique, with specimens containing straight
through cracks. It should be noted that while the activation energy
is close to that for dislocation velocity for intrinsic material,
there is a considerable shift in Tc to higher values compared with
those from the Oxford experiments. Typically the shift is -100°C
for
3
comparable slow strain-rates.
Fig. 2 also shows that the point C obtained (37~m) in intrinsic
material does not fallon material with the standard l3~m crack
depth. temperatures for a larger crack size is significant
for a larger crack-depth the line for intrinsic
This shift to higher and will be discussed in
§8.
12
II
~ c~
12
Figure 2. Plots of ~n(K) versus liTe for intrinsic (I) and n-type
(N) Si, for the Oxford experiments, and also for St.John's
experiments on intrinsic material. Point C is for intrinsic Si with
a precursor crack radius of
37~m; all other Oxford data are for a crack radius of l3~m.
4. ETCH PITTING STUDIES OF DISLOCATION DISTRIBUTION
Specimens which fractured at test temperatures from room
temperature up to only a few degrees below Tc show no significant
dislocation activity.
, '~;LF:'~~~':.::" " •• '·t ••·• .. : .. ····1 -' ... . .. ~
/'
25jJm [1I1lr lllil
[1 iOI
Figure 3. Tracing of etched fracture face of a "transition"
specimen; long rays of dislocations emanate from the crack front,
mostly from the positions
(X, Y) where the tangent to the crack front lies in a slip
plane.
However, at the transition temperature, when the specimen fractures
at a considerably higher stress than in the low temperature brittle
region (see
4
fi~. 1), the etched fracture face shows trains of dislocations
along the <110) directions, emanating from the precursor crack.
Fig. 3 shows a tracing of the optical micrograph of the etched
surface of such a specimen which failed at K=1.6 MPamt showing
these trains of dislocations extending about 100~m into the
specimen. A single dislocation train contains 80-100 dislocations,
which are approximately in the form of an inverted pile-up. Two of
the prominent rows of dislocations appear to come from points X and
Y, where two of the {111} planes inclined to the cleava~e plane are
tangential to the crack front. The rows along the third <110)
direction (normal to the surface) seem to come from near the points
on the precrack where the third intersecting (111) plane is
tangential to the crack (near the specimen surface). The fracture
surface showed no evidence for stable crack extension before
fracture.
The value of K at which dislocations begin to be emitted at Tc was
determined by prestressing the specimens at Tc to values of
K<K1c (where K1c is the low temperature fracture toughness, =
1.17 MPamt ) , and then fracturing and etching them at room
temperature. These tests show that significant dislocation activity
at the crack tip, in a constant strain-rate test, begins at a value
of K very close to K1c (-0.9Klc<K<Klc) [4].
Specimens deformed above Tc in the ductile region, up to the yield
stress, show extensive slip over the whole specimen.
5. A DYNAMIC CRACK TIP SHIELDING MODEL FOR THE BOT
The dislocation distributions revealed by etch pitting at Tc (see
figure 3) suggest that at the BOT a series of dislocation loops is
emitted from sources at or close to particular points on the curved
crack profile, where the line of intersection of the {111} glide
plane with the crack plane is tangent to the crack profile. The
screw parts of the dislocation loop cross-slip around the crack
profile, causing crack blunting, while the edge parts of the loop
move away from the crack and cause shielding. The local stress
intensity factor Ke is given by
(1)
where K is the applied stress intensity factor, and I:KD is the
shielding effect of the emitted dislocations (Thomson [5]). In the
model discussed below we have considered only the shielding
effect.
When crack propagation occurs entirely by a brittle mechanism, i.e.
bond rupture at the tip, without generation of dislocations, as
appears to be the case in the present experiments, the local
criterion for fracture in pure mode I loading is that
(2)
When plastic flow occurs during a constant strain-rate test in the
transi tion region, as the applied K increases, I:KD will increase
as the number of emitted dislocations increases, and Ke may
increase depending on the tes t conditions. Tc is then defined as
the lowes t temperature for a given strain-rate at which K1e<K1c
for all points on the crack profile, for all values of K ) K1c ' up
to some predetermined value, e.g. that at which general yielding
occurs.
5
In order to make the modelling tractable, a number of
simplifications have been made:
1. Mode I deformation is replaced by Mode III deformation. Although
Mode III calculations may not give numerically correct estimates,
they will give valuable insight into the factors important in the
dynamics of the problem. We have therefore used the fracture
criterion
(3)
2. The real, curved, crack with four sources has been replaced by a
straight crack with two sources, X, Y . The geometry is shown in
figure 4; the Burgers vector of the dislocations is parallel to the
crack front.
crack
Figure 4. Simplified model of the crack front and dislocation loops
used for the computer simulation. Loops expand from points 'Y' and
'X', eventually
to cover point 'Z'.
3. We have assumed that the velocity of the edge components of the
loop is the same as for the screws. This implies that the loops are
elongated in the screw direction (i. e. the screws are twice as
long as the edges). Thp. crack tip and dislocation interaction
stresses on the screws have then been calculated assuming them to
be straight and parallel, but a line tension term (written as a
configurational stress, i.e. force per unit Burgers vector) has
been included which takes account 'of the dislocation image stress,
and of a curvature effect in an approximate way.
4. The interaction between dislocations from different sources has
been neglected.
We now assume that the dislocation loops at the source can be
nucleated and move away from the tip provided the stress at a
critical distance Xc from the tip is sufficient to expand the loop.
Once nucleated, the back stress from this dislocation shields the
source and the stress at Xc drops below the critical value for loop
expansion. As the dislocations move away. the stress at Xc
increases again. and when the critical stress is reached another
dislocation is emitted, and the cycle repeats.
The first dislocation is emitted at a critical value of
6
(4)
- where a defines the strength of the line tension/image stress
[6].
The point Z on the crack front dislocation loops have expanded
past
(see fig. 4) is shielded only when the Z. Then the shielding at Z
is given by
K = K -ez 1: ~ j>jo (21TXj)t
(5)
summing over all the dislocations which have moved past Z. Thus
shielding at Z only starts at a time to which depends on the
distance XZ [6] .
6. RESULTS OF COMPUTATIONS FOR Si
The program simulates dislocation motion for given 'experimental'
conditions of K and T. Dislocation nucleation conditions are
specified by selection of two of KN• a and xc' with the third then
determined by equation (4). The c"alculations begin at K=KN• with
one dislocation at x=xc' Values of KN between 0.2K1c and 0.95K1c
were used. With a = 1/4 in equation (4). the former corresponds to
Xc - 10.7b. The actual values of Xc and a used to give a particular
value of KN were not found to influence the results. The
dislocation velocity data used were those of George and Champier
[2] for screw dislocations in silicon. For more details of the
method used, see [6].
Computations have been carried out for various distances dcrit=XZ,
up to 7.5~m. which corresponds to the case for a semicircular crack
with radius 13~m. used in most of the experiments.
2 ••
450 500 550 800
Figure 5. Predictions of applied K (solid lines) and extent of
dislocation array (dotted lines) at fracture. A smooth
brittle-ductile transition is predicted for KN=0.95K1c. Similar
results are predicted for 0.2Kl c <KN<O.95Klc' the transition
temperature increasing with increasing KN.
Figure 5 shows values of KeRF at which fracture occurs (i.e. when
Kez=Krc ) as a function of temperature (at K=886Nm- 3 / 2 s- 1 ,
dcrit=7.5~m and
7
KN=0.25MPamt, and the corresponding distances travelled by the
leading dislocations in the train, d~. This curve shows clearly
that the predicted transition is 'soft', i.e. the fracture stress
(proportional to KF) increases gradually with temperature, and even
below Tc (-550·C). say at 500·C, the leading dislocations would
have moved large distances and many dislocations would have been
emitted. This behaviour is contrary to that observed; in practice
no significant plasticity is detected by etching even a few degrees
below Tc ' and the curve of KF versus temperature is very sharp
(see fig. 1). Thus, although the computations predict the correct
range for Tc ' the sharp nature of the transition is not
reproduced. Further calculations have been carried out for various
values of KN, up to 0.95Krc' In all cases a soft transition is
predicted. which is not experimentally observed. The etching
experiments discussed in §4 suggest that a nucleation 'event'
occurs at a value of applied K just below Krc when dislocation
activity begins. Computations were therefore carried out in which
the calculations were started at an applied K=Ko ' at which
dislocations begin to be generated (where KN<Ko <Krc )' This
simulates a nucleation event (a possible mechanism for such a
delayed nucleation is discussed in §8).
K,(MPanl',) d F (~m) KF(MPa,J'~) dF (~m)
200 200
,00 '00
Temperalu ra (oe) Temperalure (oe)
Figure 6. Variation of BOT with Ko' Predictions of applied K (solid
lines) and extent of dislocation array (dotted lines) at fracture
for two different values of Ko ' for KN=0.2Krc ' A smooth
brittle-ductile transition is predicted for low values of Ko ' with
the transition becoming sharp as Ko
approaches Krc '
Figure 6 shows calculations of KF and dF as a function of
temperature for two values of Ko ' with KN=0.2 Krc ' These
computations show clearly that a sharp transition is predicted for
Ko-0.95Krc' consistent with the experimental observations that no
significant dislocation activity occurs at Tc until
-0.9Krc<K<Krc' The value of Tc predicted for this standard
strain rate (-535·C) is slightly below the observed Tc (-550·C),
but in view of the approximations in the model this can be
considered as good agreement.
Calculations carried out for Ko=0.95Krc' and various values of KN,
show that for values of KN«Ko the form of curves is insensitive to
KN in that a sharp transition occurs. but that as KN - Ko' the
transition tends to become soft again [6]. The sharpness of the
experimentally observed transition suggests that KN«Ko' but the
exact value of KN is not known. In most of the calculations we have
assumed KN=0.21Krc = 0.25 MPam t .
Figure 7 shows calculations of Ke z and dF as a function of time
at
8
three temperatures, for Ko=O.95K1c' KN=O.25 MPam!. At Tc (-535·C)
(figure 7(b)), Kez drops slightly initially and then increases very
gradually with time, predicting fracture at KF-2.l MPam! This
initial drop in Kez' with a longer-term rise, eventually to reach
K1c , is the reason for the steplike sharp transitions in figure 6.
At higher temperatures >Tc (figure 7(c)), the drop in Kez is
greater and Kez reaches K1c only after very long times, when
applied K is very high. (In this specimen geometry such high values
of K correspond to stresses above the overall yield stress). At
5l0·C the behaviour is totally brittle, in that dislocations do not
pass Z before K reaches K1c (figure 7(a)).
'.
'" , < •• ' .. --- -
.~
535°e 5700 e
Figure 7. Characteristics of a sharp brittle-ductile transition.
Kez is shown as a function of time for three temperatures: (a)
510°C; brittle. Kez reaches K1c before dislocations pass dcrit
(7.5pm). (b) 535°C; transition. Kez diverges from applied K exactly
at K1c (lKF) , drops rapidly and later rises to reach K1c with
applied K=2KF. (c) 570°C; ductile. KF is high; the associated
stress level is above that for general yielding. Note that an
increase in temperature from just below to just above 535°C will
produce a
jump in KF from lKF (=K1c ) to 2KF.
7. A MODEL FOR NUCLEATION OF CRACK TIP SOURCES
The computations and the experiments described above show that the
nature of the brittle-ductile transition in Si is sharp because no
significant dislocation generation takes place at the crack tip
until crack tip sources are generated at Ko just below K1c . Once
formed these sources begin to operate at KN «Ko)' sending out
avalanches of dislocations which produce rapid shielding. Assuming
that the cross-slip process is fast compared with the time taken by
the dislocation to reach the crack tip, the value of K at which the
source is formed, i.e. Ko ' is readily estimated. The stress on a
dislocation a distance r from the crack tip, in mode I loading, is
given by
l' = Kf (81Tr) t
where f is an orientation factor. Writing the dislocation velocity
vas:
(6)
9
dr A 'I'm exp(-U/kT) ,-mvo (7) V dt we find:
dr _(Kf)m
dt
At constant strain rate K (assuming f to be constant) this can be
integrated to give
m 2(m+l) (8lT)m/ 2 ro (1+ 12)
(m+2) fm
K (-) (9)
where Kd is the value of K at which the dislocation begins to move.
and ro is the distance that the dislocation travels to the crack
front. Since m varies only slowly over the temperature range of
interest. and assuming that for a given structure Kd • ro and f are
independent qf temperature. Ko is independent of t~mperature and a
function only of (K/vo )' so that for a given value of (K/vo )' Ko
is constant. This means that at Tc' assuming a constant initial
dislocation structure. the dislocation will always reach the crack
tip (and form a source) at the same value of the applied K (=Ko)'
independent of the loading rate. The condition for the
brittle-ductile transition can now be restated. namely. that a
dislocation source is formed at the crack tip just in time for the
source to emit sufficient dislocations to shield the most
vulnerable points on the crack front before K reaches KIc ·
Equation (9) also shows that Tc depends on dislocation structure.
in part~cula1+mt~rough roo Kd . Assuming Ko' Kd • f to be constant.
(vo/K) «ro / • and therefore for larger roo Vo must be greater for
a given K. and this implies an increase in Tc. Since the size of
the plastic zone scales with the size of the crack introduced by
indentation. we expect (at constant K) Tc to be larger for larger
crack sizes. as observed (see fig. 2. §3) .
From the computer calculations in §7. Ko-KIc-l.17MPam!. Kd is
likely to be determined by the dislocation loop lengths in the
plastic zone. Transmission electron micrographs of sections through
the plastic zone suggest dislocation loop lengths (.~) of the order
of a few microns (see Samuels and Roberts [4]). Assuming that the
critical stress for dislocation movement is -~b/~.
(10)
and using f-t. ~/b-l04. ro = 13.3~m. we find Kd-O.46MPamt . With
the same values of f. roo Ko-K1c ' and with m=1.2. A=1.51xl0- 4 ms-
1 (Nm- 2 )1.2 for intrinsic ma~erial (George and Champier [2]) Tc
can be calculated directly for a given K. Table 1 shows predicted
and observed values of Tc for the slowest strain rates for the two
crack sizes used. for intrinsic Si with U=2.1ev (George and
Champier [2]). The value of Kd is assumed to be the same for both
crack sizes (Kd =0. 46MPam!) . The agreement is reasonable.
particularly for the smaller crack size. For the larger crack size.
Tc is predicted to increase. in qualitative accord with experiment;
the numerical discrepancy may be due to a different value of Kd
.
10
r o (11m)
K{Pamts~l )
Tc{OC) (experimental)
37.4±1.4
628±2
598±2
To check the proposal that the sharp BDT in Si is controlled by
existing dislocations in the crystal (namely those in the plastic
zone of the indentation) moving to the crack tip where they
generate new sources which shield the crack, the top 41lm of the
precracked surface of intrinsic Si specimens were polished away,
thus removing much of the plastic zone at the surface. Figure 8
compares the BDT from such a specimen {curve (b)) with that of
un~olished specimens {curve (a)). It is clear that Tc increases by
-55°C at K-890Pamts- 1 • This increase confirms the importance of
the existing dislocations in the plastic zone in the specimens, and
this suggests that the transition is still controlled by nucleation
of crack tip sources by dislocations which have not been removed by
the polishing treatment. The increase in Tc then implies that
existing dislocations either have a higher operating stress/smaller
loop length (i.e. greater Kd )
or are further away from the crack. (For further details, see
Warren [7]). The higher values of Tc found by St.John [3] in his
experiments, compared to those in the Oxford experiments, for the
same. strain rates (see fig. 2, §3), are also attributed to a
smaller dislocation density/source size in St.John's experiments.
In the latter's experiments the cracks were introduced in a
different manner, without forming a surface plastic zone. The
origin of the dislocation sources in his crystals is not
known.
(al c) (b) • Br1ttIe
!600
540
Figure 8. Failure stress versus temperature for: (a) 'Control'
specimens. Tc=545°C. (b) 'Unabraded' specimens. The top 41lm of the
tensile surface was removed by chemo-mechanical polishing.
Tc=595°C. (c) 'Abraded' specimens. The top 41lm of the tensile
surface was removed by chemo-mechanical polishing. The surface was
then abraded with 61lm diamond paste. Tc=555°C.
11
A further check of the model has been made by grinding the surface
of the polished specimen. This should introduce dislocations at the
surface, and Tc would be expected to decrease again. Fig. 8(c)
shows the experimental results; Tc is now 40°C lower than for the
polished specimen, in agreement with the predicted trend.
The computer simulations of the emission of dislocations from the
crack tip predict that a sharp transition occurs when crack tip
sources are formed at Ko-K1c ' which, once formed, operate at KN«Ko
(see fig. 6). If, however, such sources already exist in the
precracked specimen before the test begins, the transition should
be "soft", as shown in fig. 5. No nucleation is necessary in this
case before efficient shielding takes place. To test
700
o 1lriItie. PI'_formed • 1kittIe. """,,01 • Ductile. """"01
• Predicled ........ ,.;'",
/):. ' P ' . ' ,
2
Figure 9. Failure stress versus temperature for pre-deformed
specimens. Also shown for comparison are the experimental results
from fig. 8(a), and
the computer simulation curve for KF versus temperature, for
KN=O.2K1c .
this prediction, standard precracked specimens of intrinsic Si were
deformed at Tc to a value of K-O.9K1c' then unloaded, cooled to
lower temperatures and reloaded to fracture at the new temperature.
The pre-deformation should be just sufficient to nucleate crack tip
sources, or to make the time needed for nucleation negligible. Fig.
9 shows the results of such a set of experiments. The BOT of the
pre-deformed specimens is now "soft" as predicted, and crack tip
plasticity induced at lower temperatures. Fig. 9 also shows for
comparison a computer simulation curve for the case KN=O.2K1c ' for
comparable strain rates. The experimental curve is even softer than
that predicted for this particular value of KN. Observations of
dislocation distributions on the etched surface of the cracked
specimens show a progressive increase in dislocation generation
with increasing temperature of testing, as expected. (For further
details, see Warren [7].
7. SUMMARY
In pre-cracked low dislocation density Si the BOT is controlled by
existing dislocations. A sharp transition occurs because, at the
transition temperature, crack tip sources are not nucleated until
the applied K is very close to K1c . These sources then operate at
a much smaller K=KN. The strain rate dependence of Tc is controlled
by the activation energy for dislocation motion, not the activation
energy for loop nucleation at the crack-tip as proposed~ Rice and
Thomson [8]. The BOT in Si occurs at the
12
lowest temperature at which crack tip sources are nucleated below
Krc and the emitted dislocations shield the most vulnerable point
of the crack quickly. enough so that Ke z <Kr c for K values up
to those for macroscopic yield.
The model for the formation of crack tip sources predicts that Tc
should decrease with increasing dislocation density, and that for
large dislocation densities, nucleation of crack tip sources is not
necessary, and the resultant transition will be soft. This may be
the case for pre-cracked specimens of b.c.c. metals, where soft
transitions have been observed (Hull, Beardmore and Valintine
[9]).
The main parameter controlling the BOT is the dislocation velocity
and any mechanism which reduces the average velocity (such as
radiation damage, or precipitation hardening) is likely to increase
Tc'
Acknowledgements
Our thanks are due to the S.E.R.C. and B.P. Venture Research Unit
for financial support.
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
REFERENCES
Newman, J.C. Jr. and Raju, I.S., An factor equation for the surface
crack. 185-192.
empirical stress-intensity Eng.Frac.Mech., 1981, 15,
George, A. and Champier, G., Velocities of screw and 600
dislocations in n- and p-type silicon. Phys.Stat.Sol., 1979, 53a,
529-540. St.John, C., The brittle-to-ductile transition in
pre-cleaved silicon single crystals. Phil.Mag., 1975, 32,
1193-1212. Samuels, J. and Roberts, S.G., The brittle-to-ductile
transition in silicon. I. Experiments. Proc.R.Soc.Lond., in press.
Thomson, R., The physics of fracture. In Solid State Physics, ed.
H. Ehrenreich & D. Turnbull, Academic Press, New York, 1986,
pp.1-129. Hirsch, P.B., Roberts, S.G. and Samuels, J., The
brittle-to-ductile transition in Silicon. II. Interpretation.
Proc.R.Soc.Lond. , in press. Warren, P.O., The brittle-to-ductile
transition in silicon: the influence of pre-existing dislocation
arrangements. Scripta Metall., in press. Rice, J .R. and Thomson,
R., Ductile versus brittle behaviour of crystals. Phil.Mag.,
1974,29, 73-97. Hull, D., Beardmore, P. and Valintine, A.P., Crack
propagation in single crystals of tungsten. Phil.Mag., 1965, 12,
1021-1041.
13
R A AINSWORTH
ABSTRACT
Under steady loading, the stress and strain rate fields near the
tip of a stationary crack relax from high initial values, describe~
by the parameter C(t), to steady state values described by the
parameter C. For creep ductile materials, the transitional phase
before attainment of the steady state is usually neglected and
crack initiation and s~bsequent creep crack growth rates can be
determined from calculations of C. However, for creep brittle
materials it is important to estimate the additional strains
accumulated near the crack tip during the period of stress
redistribution.
The steady state amplitude C* may be estimated for cracks in
components using approximate reference stress methods. In this
paper an estimation formula for C(t) is developed and expressed in
these reference stress terms. The formula is particularly
convenient for integration to obtain strains near the crack tip.
This integration is performed and used to assess the effect of the
initial period of stress redistribution on creep crack initiation
and growth. It is shown that the transitional effects may be
a~proximat.ely described by the factor h+ elastic strain at the
refer ence stress/creep strain at the reference -stress]. For
creep ductile materials this factor will often be close to unity,
but for creep brittle materials where the accumulated creep strains
may be low, it can be signi ficantly greater.
INTRODUCTION
Riedel and Rice [1] have shown that for a cracked body deforming by
creep,
the near-tip stress and strain rate fields are of the HRR type
[2,3];
14
i.e. for polar co-ordinates (r,e) centred at the crack tip
[C(t)/Blnr]l/(n+l) °ij(e,n)
B[C(t)/Blnr]n/(n+l) Eij(e,n)
hold for r + o. Here, Band n are constants in the uniaxial creep
law
.c E:
(1)
(2)
(3)
In' 0ij and E:ij are known dimensionless functions of the creep
index n, and
the influence of load, geometry and time on the near-tip fields
is
described by the amplitude C(t). For a body subjected to steady
load,
steady state creep conditions hold at long times and
* C(t) + C as t + ~ (4)
* Finite-element solutions for the steady state amplitude, C ,
are
available in [4] for a variety of crack sizes, geometries and
stress
indices. For practical applications these solutions may be
approximately
described by the formula [5,6]
* .c C °ref E: ref R (5)
where °ref Po IPL( 0 ,a) (6) y y
R = K2/02 (7) ref
Here PL is the limit load for a yield stress 0y for the cracked
geometry
with a crack size a, and K is the elastic stress intensity factor.
The .c
strain rate E:ref is determined from uniaxial creep data at the
stress level
o f; for example from eqn (3). The practical significance of eqn
(5) is re *
that it enables C to be calculated readily as stress intensity
factor and
limit load solutions are widely available (for example, [7,S]), and
it
enables realistic creep strain rate data to be used rather than
being
restricted to power law creep.
15
In addition to the long-time behaviour of C(t) being known by eqn
(4),
the response at short times is also known. For a body which
behaves
elastically on first loading
C(t) + K2/(n+1)E't as t + 0 (8)
where E' = E, Young's modulus, in plane stress and E' = E/(1-v2),
where v
is Poisson's ratio, in plane strain [1]. At intermediate times,
simple
addition of the short-time and long-time limits
C(t) * * C [1 + K2/(n+1)E'C t] (9)
has been found to be a good fit to numerical results for simple
structures
under constant load [9]. Unfortunately, the lIt factor in eqn (9)
is not
particularly convenient for obtaining the near-tip strains by
integration
of eqn (2). In this paper, a new estimation formula for C(t)
at
intermediate times is developed from an estimate of the J-integra1
in the
far-field.
ESTIMATION POKKDLA POR C(t)
* The parameter C may be, defined by a path-independent line
integral which
is similar to the J-integra1. The similarity ensures that in
steady-state
creep, J increases at the rate
dJ/dt * C as t + ~ (10)
For a body which is loaded elastically at time t = 0,
J (11)
Rather than assume that C(t) could be obtained from summation of
the 10ng
and short- time solutions of eqns (4) and (8), Ainsworth and Budden
[10]
have recently estimated C(t) by assuming that J could be
approximately
obtained by summation of the long- and short- time solutions of
eqns (10)
and (11) as
This approximation was confirmed by finite-element analysis of a
compact
tension specimen [11].
The near-tip stress and strain fields are related to the J-integra1
in
a similar manner to the relationship between C(t) and the stress
and strain
rate fields in eqns (1) and (2). This enables C(t) to be evaluated
from an
estimate of J, and omitting mathematical details the J estimate in
eqn (12)
is consistent with a C(t) estimate [10]
C(t) C* {(I + E'C*t/K2)n+1
(1 + E'C*t/K2)n+1 - 1
For t ~ 0 and t ~ ~, eqn (13) reduces to eqn (8) and eqn (4),
respectively.
x ~ eqn (9)
----x----x----x----x----x _____
(13)
Figure 1. Variation of C(t) with time under steady load.
17
Equation (13) is compared with eqn (9) and the finite-element
results
of [11] in Figure 1. It is apparent that while both the
estimation
formulae are close to the finite-element solution, eqn (13)
provides the
better approximation. It is also apparent that C(t) rises to values
very
'* much higher than the steady state value C and that it is,
therefore,
important to assess the effect of the transient phase in Figure
1.
From examination of Figure 1 and eqns (9) and (13), it is
convenient
to introduce a redistribution time, t red
(14 )
Equation (13) suggests that C(t) is within 10% of the steady state
value,
'* C , for t ) tred for n ) 2.5. The redistribution time may be
written more
conveniently by combining eqns (5,7,14):
(J IEec ref ref (15)
The redistribution time of eqn (15) may be interpreted as the time
for
which the accumulated creep strain at the reference stress becomes
equal to
the elastic strain at the reference stress. This interpretation may
be
used to extend the definition of redistribution time to incorporate
primary
creep strain as
(16)
where the left-hand-side of eqn (16) is the accumulated creep
strain at the
reference stress level.
In the next section, models for creep crack initiation and growth
are
discussed which involve the accumulated creep strain near the crack
tip.
As creep strain rate is given by eqn (2), these models require a
value for
the integral of C(t) to the power n/(n+l). It transpires that eqn
(15) is
particularly convenient for this integration and leads to the
closed-form
expression
c'* n/(n+l) [(1 + tit )n+l _ l]l/(n+l) t . red red (17)
18
2
Figure 2. Variation of near-tip creep strains with time.
where t red is defined by eqn (14). Equation (17) is shown as a
function of
time in Figure 2 for n = 5. It is apparent that the integral can
be
closely approximated as an initial increment plus a steady state
value
f t C(t)n/(n+l)dt ~ C* n/(n+l) t rl + t ttl o . red- (18)
Equation (18) may be expressed in reference stress terms using eqn
(15) to
give
ft C(t)n/(n+l)dt ~ C* n/(n+l) t r(1 + a f/EEc f(t)] o . re re
(19)
It can be seen that the creep strains near the crack tip are
increased
above those which would have been accumulated under steady state
creep by
the factor [1 + elastic strain at the reference stress/creep strain
at the
reference stress].
CREEP CRACK INITIATION AND GROWTH
Prior to crack growth there is often a period of crack initiation
which
involves blunting of the crack tip without significant crack
extension.
The blunting process has been addressed in r12] for steady state
creep
conditions and may be represented by an increase in crack opening
displace-
ment, 0, with time. The rate of crack opening, 0, is associated
with
strain rates on the blunting crack tip by
.c e: (0/0) f (e,n) (20)
where the strain rates depend on angular position, e, around the
notch and
also on the stress index, n, according to the function f(e,n) which
is
known approximately from analyses of crack tip blunting [13].
Following
the steady state analysis in [12] but allowing for the transitional
phase
discussed above, the line integral definition of C(t) enables the
crack
opening displacement to be derived as a function of time:
o(t) 1/ t
(21)
where the crack opening displacement has been assumed to be zero on
initial
loading and h(n) incorporates angular integration of f(8,n) and
some
constants of integration. It transpires that h(n) ~ 1, sensibly
indepen
dent of n. Substituting eqn (19) into eqn (21) and using eqn (5)
to
* esti~~te C leads to the simple expression
[O(t)/R]n/(n+l) e:c f(t) + C1 fiE re re (22)
If crack growth is assumed to start when the crack opening
displace
ment reaches a critical value, 0i' then the initiation time follows
from
(0 /R)n/(n+l) i (23)
The left-hand-side of eqn (23) may be considered to define a
"critical
strain" for crack initiation which depends on both 0i and on
geometry
through the length parameter R of eqn (7). For creep brittle
materials
with low values of 0i' the elastic strain (C1ref /E) contributed by
the
20
transitional phase can be a significant contribution to the
critical
strain. Indeed, in extreme cases the critical strain can be less
than the
elastic strain at the reference stress and crack growth must be
assumed to
start at time t = o.
Experimental creep crack growth rate data subsequent to initiation
are
* often found to correlate with the parameter C according to
(24)
where A and q are experimentally determined constants. In the
transitional
phase before attainment of steady state conditions, eqn (24) may
be
* expected to apply but with C replaced by C(t). The constant q is
found to
be close to unity and sometimes taken as n/(n+l) so that crack
growth is
consistent with critical strain models of failure. Under these
circum
stances the integrated growth becomes (see also [14])
6a(t) = aCt) - a(o) n/(n+l)
AC* t [1+0 flEec f(t)] re re (25)
Thus the crack growth is equal to the steady state crack growth
times the
factor [1+0ref/Ee~eft]. This may be generalised approximately for
the case
q*n/(n+l) as
(26)
Clearly, when combined initiation and growth is being considered,
it is not
necessary to include the additional strain 0 fIE into both
calculations. . re Thus the results of this section show that
stress redistribution
effects on both crack initiation and growth may be incorporated
by
evaluating the factor [1+0 flEec f]. If creep strains greatly
exceed re re elastic strains, as they often do for ductile steels,
then steady state
solutions are applicable and redistribution effects can be
neglected. More
generally, however, redistribution effects lead to earlier crack
initiation
or increased crack growth and these can readily be assessed in
terms of the
simple factor.
Acknowledgement: This paper is published by permission of the
Central
Electricity Generating Board.
21
REFERENCES
1. Riedel, H. and Rice, J.R., Tensile cracks in creeping solids, in
Fracture Mechanics: Twelfth Conference, ASTM STP 700, 1980, pp
112-130.
2. Hutchinson, J.W., Singular behaviour at the end of a tensile
crack in a hardening material, ~. Mech. Phys Solids, 1968, li, pp
13-31.
3. Rice, J.R. and Rosengren, G.F., Plane strain deformation near a
crack tip in a power law hardening material, ~. Mech. Phys. Solids,
1968, li, pp 1-12.
4. Kumar, V., German, M.D. and Shih, C.F., An engineering approach
for elastic-plastic fracture analysis, EPRI Report NP-1931,
1981.
5. Ainsworth, R.A., Some observations on creep crack growth, Int.
J. Fracture, 1982, 20, pp 147-159.
6. Miller, A.G. and Ainsworth, R.A., Consistency of numerical
results for power law hardening materials, and the accuracy of the
reference stress approximation for J, Engng. Fract. Mech. in
press.
7. Tada, H., Paris, P.C. and Irwin, G.R., The Stress Analysis of
Cracks Handbook, 2nd edn, Del Research Corp., St. Louis,
Missouri,-r9~
8. Miller, A.G., Review, of limit loads of structures containing
defects, Int. J. Pres. Ves. and Piping, 1988, ~, pp 197-327.
9. Ehlers, R. and Riedel, H., A finite element analysis of creep
deformation in a specimen containing a macroscopic crack,
Proceedings of Fifth International Conference on Fracture, ed D
Francois, Pergamon, Oxford, 1981, Vol 2, pp 691-298.
10. Ainsworth, R.A. and Budden, P.J., Crack tip fields under
non-steady creep conditions, I : estimates of the amplitude of the
fields, CEGB Report RD/B/6005/R88, 1988.
11. Budden, P.J. and Ainsworth, R.A., A finite-element analysis of
crack tip fields under non-steady creep conditions, CEGB Report
RD/B/6038/R88, 1988.
12. Ainsworth, R.A., The initiation of creep crack growth, Int J.
Solids Structures, 1982, ~, pp 873-881.
13. Ainsworth, R.A. Approximate blunting solutions for tensile
cracks, Applied Solids Mechanics - 1, Elsevier, London, 1986, pp
59-72.
14. Ainsworth, R.A. and Budden, P.J., Crack tip fields under
non-steady creep conditions, II: estimates of associated crack
growth, CEGB Report RD/B/6006/R88, 1988.
22
Operational Engineering Division (South) Canal Road, Gravesend,
Kent DA12 2RS, UK
ABSTRACT
Of the parameters which characterise conditions at the crack tip,
those which can be accurately calculated are contour integrals
around the tip. For a viscous material an integral C* is most
appropriate. For solids which creep, other integrals, which are
valid in the non-steady state case, have more physical
significance, may be more readily determined from stress analysis
results, and for steady state secondary creep tend to a value very
close to that of C*. To find out which integral provides the best
correlation for ~CrMoV Steel several such integrals are computed
for compact tension specimens of this material, under both load and
displacement controlled loading using data on crack growth at
838K.
INTRODUCTION
At high temperatures, methods are being developed to predict
failure of
materials in service conditions from measurements in test
conditions. A
comprehensive text on the topic has recently been produced by
Riedel [1]. Failure occurs from the growth of either cavities or
cracks.
In the latter case, the most appropriate parameters for
correlating
between service and test conditions, both the initiation of the
growth of
a crack and its subsequent rate of growth, are contour integrals
around
the tip. Provided that there is a single parameter stress field at
the
tip, Riedel [1] indicates those regimes in a reference stress-time
space,
where three such integrals J, C* and a generalisation of C* which
he
denotes by C*h are adequate approximations. For constant loading, J
or
C* have also been recommended by Ainsworth, Chell, Coleman,
Goodall,
Gooch, Haigh, Kimmins and Neate [2] for the extreme cases of fast
crack
growth at small times, or slow crack growth at long times
respectively.
23
For more complex conditions such as displacement controlled
loading, the
above contour integrals can be more path dependent, so that it
is
necessary to use a contour integral near the tip and evaluate
it
numerically. Hellen and Blackburn [3] have recently reviewed
the
calculation of contour integrals around crack tips when they are
not
necessarily path independent.
The purposes of this investigation are to indicate how s~ch
contour
integrals may be used to characterise crack tip conditions under
creep,
and to present the methods available to calculate them from the
results
of finite element calculations. The finite element program BERSAFE
[4]
calculates stresses, strains and displacements for materials
undergoing
elastic, plastic and creep deformations. The post-processing
program
PLOPPER [5] can then evaluate these integrals. This can be done
directly
for elasticity and plasticity. The methods for creep are
illustrated in
subsequent sections for two dimensional geometries. Axisymmetric
and
three dimensional geometries may be treated similarly.
These methods are applied to creep crack growth data obtained
by
Neate [6] from both load and displacement controlled compact
tension
specimens of ~CrMoV steel at 838K.
CONTOUR INTEGRALS IN TVO DIMENSIONS INCLUDING CREEP
Far field integrals may be calculated most easily, but to be sure
that
they adequately describe what is happening at the tip, they should
be
determined from a contour around the process zone. To obtain
satisfactory numerical accuracy, this is done by evaluating a
contour
int~gral ona contour well away from the tip and correcting it to
its
value on a contour around the crack tip elements by adding a
surface
integral over the area between the contours. The outer contour
of
integration is fixed and includes the crack tip throughout its
growth.
For the general situation there are four incremental integrals
over
a time At which can be calculated by PLOPPER for a far field
contour and
a corresponding 4 integrals which can be related to a near tip
contour by
adjusting the first 4 integrals by adding a surface integral over
the
area in between, as explained by Hellen and Blackburn [3]. The
contour
integrals can be written in incremental form as:
24
(_J _ Bji»Nk-l:J.(Ti ax~)} ds (1) k ax.
~
au. l:J.J* - l:J.T* - l:J.T* f{l:J.WNk - l:J. (T. ~)} ds (2) wk k
pk ~ xk
au. al:J.u. l:J.Tbk f{!o t .. l:J.(_J _ Bj i) Nk - T. ax ~} ds
(3)
~J ax. ~ ~ k
f{l:J.WNk - a (4) l:J.Tck T. a l:J.u.} ds ~ xk ~
PLOPPER can provide values for each of these integrals, both
around
the chosen outer contour and also with an adjustment for the
appropriate
surface integral. To evaluate the integrals defined by equations
(1) and
(2) J*wk and J*k are calculated by PLOPPER, and then the difference
is
taken of the values at times l:J.t apart. This difference can then
be
divided by l:J.t to produce a rate of change of contour integral.
To
evaluate the third and fourth integrals, the program PREPLOP is
used l:J.W l:J.U i
before PLOPPER to change Wand ui to l:J.t and ~, and also all
components
of strains to their time derivatives. PLOPPER then evaluates the
ratios
of the third and fourth integrals to l:J.t. PLOPPER can now
evaluate a
further integral ~ which can be defined in certain circumstances
which
will be referred to in the next section. Brust and Atluri [7]
computed
the second and fourth of these integrals, but concluded that there
is
insufficient experimental data to say which of these integrals
best
correlates the rate of crack growth.
Stationary Cracks
For a stationary crack, Riedel and Rice [8] and Hawk and Bassani
[9] have
shown that creep will dominate elasticity near the tip, and
have
determined the tip asymptotic field assuming path independence of
the l:J.T
contour integral l:J.tc . When elasticity is negligible, this
approximately
tends to the parameter C*h of Riedel [1]. For secondary creep
C*h
simplifies in turn to the integral C* defined for a viscous
material by
a2u. Landes and Begley [10] as C* = J{WN1 - Nitij aX1~t} ds where W
is such
that the t .. ~J
strain rate
aw --2---' When Norton's law is obeyed, i.e. the creep a u.
3 __ J_ 3x.dt
~
is proportional to the nth power of the deviation of the
25
stress from a hydrostatic stress, Riedel and Rice [8] have shown
that the
stress and strain rate near the tip are asymptotically proportional
to r-1/(n+1) and r-n/(n+1) respectively, so that the integrands in
the
preceding integrals have an r-1 singularity, giving rise to a
finite
value for the contour integral around the tip. More
precisely,
Stonesifer and Atluri [11,12] showed that under steady state
conditions t.T t.J*
of constant static load with negligible 1 .. c d w dt e ast~c~ty,
xr- an ~ ten 0
. a2u. f{(1 +1/n) WN1 - Nitij aX1~t}dS.
By integrating around a circular contour centred at the crack
tip,
they found also that for values of n between 3 and 13, this varied
from
O.98C* to 1.00C* for plane strain and from 1.11C* to 1.14C* for
plane 2
t.Tb t.J* 1 a uj stress. Similarly, xr- and xr- tend to J{~(1+n)
iN1 - Nitijax1at} ds,
which for plane strain will be 1.02C* for n = 3. An integral C*,
which
would tend to C* in the case of Norton's law for contours within
the
region where the stresses are dominated by this singularity has
been dE ..
[ ] f l) C suggested by Bassani and McClintock 13. They replaced W
by tijd ~
where E .. are the components of creep strain rather than total
strain, ~)C dE ..
and the integration is for a specific relationship between t ij and
dt~)C, rather than by integrating in the material the rate of work
density
~~ over the actual history that each point in the material has
seen.
Stonesifer and Atluri [11,12] showed, that even for a material
obeying
Norton's law, under non-steady state conditions C* is path
dependent in dT
contrast to dtC which was path independent because it contained a
surface
area adjustment. In evaluating C* in the present investigation,
a
corresponding surface area adjustment has therefore been
included.
Brunet and Boyer [13] have also obtained similar results and noted
the t.T c advantage of xr- over C* prior to the attainment of
steady state
conditions.
For a stationary crack, the purpose of an investigation is
usually
to find the initiation of crack growth. For mode I this has
been
approximately correlated with the attainment of a critical value of
J*
[14] or of f~ C*dt [2], which may be approximated by Tc [7]. When
the
load changes, or is first applied, it should be much more accurate
to
evaluate a contour integral directly, than as the integral over
time of a
contour integral with an integrand involving time derivatives.
Riedel
26
[15] has investigated C* under a step change in load on a
material
obeying Norton's law. He found a t- 1 singularity in C* leading to
a
logarithmic singularity in the time integrated value of C*. However
by
taking the increase to be linear over a small time step be obtained
a
large value which depended on the magnitude of the time step. In
a
further investigation [1] he noted that for strain hardening
primary
creep, his generalisation C*h (which is a coefficient of the
singularity
in stress near the tip rather than a contour integral in itself)
is
singular in time at a step change in load, but that this
singularity is
integrable.
These difficulties associated with C* are likely to occur with T
c
and Tb also, hence better numerical accuracy can be obtained by
basing
the initiation of growth criteria on J*w or J*.
Growing Cracks
For a growing crack without dynamic effects, Hui and Riedel [17]
considered the case when the creep strain rate was isochoric
and
proportional to the nth power of the stress (Norton's law). For
n<3
elasticity dominates and there are no steady state solutions. For
n>3
the singularity in stress near the tip for a growth rate ~~ is
1
n-1 proportional to (~ da)
r dt As this gives rise to an infinite
contribution to the rate of change of the contour integral during
stable
growth, it may be necessary that n tends to infinity for large
strain
rates. However, Ainsworth [18] and Riedel [1] noted that the
region
where this solution is appropriate is often so small that, for most
of - 1
the life, it is determined by the surrounding
singularity in stress. The contour integrals
evaluated in this field. Riedel [1] proposed
-!. n+l field with an r 2 or r
* bJ w1 or bT1c may be an approximate formula in
terms of the initial value of J 1 , the time t and the steady
state
secondary creep value C*l' for the case of a steady load. If
the
integrals were to be evaluated within the crack tip region for a
material
creeping according to Norton's law, cr, bJ*wland bT1c would be
path
dependent, including a term which varies as £-2/(n-l) where £ is
the
minimum distance of the contour from the tip.
Theoretical Comparison of Contour Integrals
Moran and Shih [19] have recently expressed their preference for
the use
of C* as a correlating parameter on the grounds that it is a
path
27
independent contour integral for the case of secondary creep in the
, h h -1/(n+1) d' h' reg10n were t e stresses vary as r accor 1ng
to t e R1ce-
Rosengren-Hutchinson distribution (HRR) , Under other conditions
and for
other contour integrals the shape of the inner contour on which
the
integral is defined can slightly affect the value of the integral,
but
this should not matter provided the same shape is used in analysis
of
test data and in prediction calculations. This corresponds to the
inner
contour defining the inner boundary of the area over which the
area
adjustment is made to the value of the contour integral on the
contour
away from the crack tip, when the crack has grown significantly in
a
finite element investigation. A suitable shape for the inner
contour for
analysis investigations is two straight lines a small distance to
each
side of the crack and extending well away from the tip. For
this
contour, the first term in each of the integrands (ie, that
involving W, dT
does not contribute to the value of the integral. c Hence, ~, I1W,
etc)
dTb ~ and c* will be identical for such a contour, which
corresponds to a
flat process zone at the crack tip as it tends to zero. A suitable
shape
for the inner contour for numerical investigations is the Dugdale
contour
[7] which is similar to the previous shape except that, instead
of
extending well ahead of the crack tip, the parallel lines are
joined by a
semicircle or semisquare ahead of the crack. For such a contour,
little dTb dTc
difference would be expected between the values of ~, ~ and C*
as
long as the semicircle (or semisquare for a finite element
analysis) was
within the HRR zone so that the results of Stonesifer and Athuri
[11,12]
are applicable.
dT dT The relative merits of d~' C* and d~ as crack tip
characterising
parameters appear to be as follows: A. C* is independent of the
shape of
the inner contour for secondary creep. B. For inner contours
parallel to
the crack all the above parameters are identical. C. For other
inner
contours such as the Dugdale contour used in finite element
investigations, C* has a greater likelihood of loss of accuracy
because dT
of the extra numerical manipulations while d~ has a physical
significance, being the variation with respect to crack tip
position of
the difference between the rate of energy dissipation within
the
contour.
28
IKPLKKKNTATION AND APPLICATION
For a growing crack, in order to calculate the integrals near the
tip,
methods are required for updating the stresses and strains
between
successive stress analysis solutions to take account of the
changing
position of the crack tip. This can be done, either by using a
procedure
such as BERCRAG3 [20], or by choosing the mesh so that there is a
node of
the initial mesh at every position where the tip is going to be
when the
stresses and displacements are recomputed. Tests have been carried
out
by Neate [6] for a coarse grained bainitic ~CrMoV steel under
such
conditions. Norton's law should be modified to take account of
primary
and tertiary creep, but, for nominally similar material, the
measured
rates of crack growth and of change of load differed by orders
of
magnitude. Hence, in view of the scatter in the data and the
lengthy
times required for convergence, the uniaxial creep strain rate EC
was
taken to be given by Norton's law in the form, E = O.B 10-22a7/hr
when a c
is in MPa. Poisson's ratio was taken to be 0.3 and Young's modulus
to be
170,000MPa.
For half of the compact tension specimens, a finite element mesh of
139
elements with vertex and mid-edge modes (Figure 1) was generated;
the
mesh size near the tip being 0.2mm. The initial crack size was 12mm
in a
specimen of width 40mm between load line and back face and
thickness
20mm. Special elements to take account of the linear elastic
singularity
were used at the crack tip. The contour integration for the near
tip
parameters was around the inner thickened contour in Figure 1,
three
sides of an Bmm x 12mm rectangle commencing in the plane of
symmetry 6mm
ahead of the initial position of the crack tip, and finishing on
the
crack face 2mm from the initial position of the tip. The
contour
integration for far field parameters was around the outer
thickened
contour for displacement controlled loading, but around the
inner
thickened contour for load controlled loading as the outer
contour
contains the load point. In each case the tolerance was 10%.
The
displacements were imposed on the nodes on the front of the
specimen
(16mm from the load line. The actual specimen extended only 10mm,
but
Neate's measurements were at a distance of 16mm, so the mesh was
enlarged
to take account of this). The runs were done in engineering
plane
strain. A rerun of one case under mathematical plane strain made
little
difference.
29
Results are presented in Table 1 corresponding to imposed
displacements
of 0.13, 0.16 (twice), 0.19, 0.26 and 0.32mm . The cracks were
grown
between each of up to 6 BERSAFE solutions by amounts corresponding
to
those amounts measured by Neate [6]. Crack growth was taken to
commence
at 0.01 hours. The calculated loads at the location where the
displacement was measured are less than those measured by Neate [6]
where
the ~isplacement was applied, because the load on the remaining
ligament
of a compact tension specimen is mainly bending. The measured
values are
in parentheses. In four of the five cases the measured and
calculated
load relaxation rates are comparable. Also included in parenthesis
are
estimates of C* derived by Neate [6] from these relaxation rates.
In
Table 2, corresponding analyses are presented for three tests by
Neate
[6] for this material and geometry under constant load. Also
included
are the displacement 16mm from the load line, plus (in parentheses)
the
sum of the calculated initial plus measured additional displacement
at
this location, and, also in parentheses, Neate [6] estimates of C*
from
the rate of change of these displacements. C* has also been
computed
both in the far field, and, by use of a surface integral
adjustment, near
to the tip, for the three load controlled cases and for one of
the
displacement controlled cases.
N I"
O O
LL ED
W A
P~ ) ( ~ ) (~
.0
0 ., <J
• 0 ~ (near Upl J. e EaUmala 01 C*from t ~ 8 e
0
• 0 load ,ataxallon rale 0
10 or load IIna ,. 0 0 dlaplacament rat. 0 0 <J
B a 8 010
• .!J Q o,p~ 0 Q. :::E
8 a oJ 10-1 ~ iB « a: • " 0
0·00
'" • ~ Q O. ::!: " OX' a: g :::) v 0 0 oe ~ Z 10-2 0 0 0 0
• "0 o. ~ a
~T -and C* a •• function of ~I
6a - (crack growth ratel 61
FIGURE 2
33
DISCUSSION
Riedel [1] has noted that for obtaining good correlations of crack
growth
rate with other parameters, a measured parameter such as C* gives
a
closer correlation as it deals with that particular specimen of
material.
However, this does not apply for prediction purposes when only
typical
material pro~erties are known. For the load controlled cases
the
integrals ~~ and ~~*, when evaluated with the surface area
contribution,
appear to give good correlation with crack growth rate. However,
both of
these parameters have opposite signs for load and displacement
controlled
loadings. From Table 1 it is seen that the crack tip parameters
which
appear to give the most consistent correlations with the crack
growth IlT -* IlT
rate in displacement controlled conditions are Iltb ,C and IltC
defined
near the crack tip (i.e. around the 'Dugdale' contour). For
negligible
elasticity, so that C* is defined, C* lies between the IlTb
values bt and IlT IltC in the absence of crack growth.
IlT When, under fixed displacements, the
far field value of IltC is calculated near the outer boundary, its
value
was found to be zero as expected since the far field displacement
should
not be appreciably time dependent. A similarly consistent
correlation is IlJ w -obtained with bt or C* defined near the outer
boundary, but as these are
not crack tip parameters they will not be appropriate for more
general
conditions. In fact both of these have opposite signs in the cases
of
load controlled and of displacement controlled crack growth
respectively.
IlTc IlTb A log-log plot of the values of C*, Ilt and bt is
presented in Fig~re 2
and is considered satisfactory as the material's scatter varies
over two
orders of magnitude for nominally identical material under the
same
displacement. On this plot, IlTb/llt correlates linearly with the
rate of
crack growth over the full range from 0.8~m/hr to 1Smm/hr.
Estimates
are also included of C* as derived by Neate [6] from the load
relaxation
rate measured from each individual specimen. They have a similar
slope IlTc IlTb
to the computed results for bt and bt' but are typically about
a
quarter the size. (This compares with specimen to specimen
variations of
about 10 in relaxation time and about 100 in growth rate). The
good
correlations of crack rate with both the estimated parameter
(taking
account of the individually measured relaxation rate) and the
computed
parameters (which in this investigation used only standard Norton
law
34
secondary creep data) is thus less sensitive to the occurrence of
primary
and tertiary creep than might have been expected. Hence the
computed
parameters may be used to predict the order of magnitude estimates
of
crack growth rate in other conditions.
ACKNOWLEDGEMENTS
The writer is grateful to Dr G J Neate for providing fuller
information
than was published on some of his test results and to Drs T K
Hellen and
R A Ainsworth for comments on the initial draft. This paper is
published
by permission of the Central Electricity Generating Board.
REFERENCES
1. Riedel H. Fracture at high temperatures. Springer-Verlag,
Berlin, 1987.
2. Ainsworth R A, Chell G G, Coleman M C, Goodall I W, Gooch D J,
Haigh J A, Kimmins STand Neate G J. Assessment procedure for
defects in plant operating in the creep range, 1986,
CEGB/TPRD/B/0784/R86.
3. Hellen T K and Blackburn W S. Non-linear fracture mechanics and
finite elements. Engineering Computations, 1987, 4, 2-14.
4. Hellen T K and Harper P G. BERSAFE, Volume 3, Users guide to
BERSAFE, Phase III, Level 3, CEGB 1985.
5. Moyser G and Hellen T K. BERSAFE Volume 8. Users guide to
PLOPPER Level 3, CEGB 1985.
6. Neate G J. Creep crack growth in ~CrMoV steel at 838K. Mat Sci
Engng, 1986, 82, 59-84.
7. Brust F Wand Atluri S N. Studies on creep crack growth using the
T* integral. Engng Fracture Mechanics, 1986, 23, 551-574.
8. Riedel H and Rice J R. Tensile cracks in creeping solids.
Fracture Mechanics ASTM STP700, 1980, 112-130.
9. Hawk D E and Bassani J L. Transient crack growth under creep
conditions. Jnl Mech Phys Solids, 1986, 34, 191-212.
10. Landes J D and Begley J A. Mechanics of crack growth, ASTM
STP590, 1976, 128-143.
11. Stonesifer R Band Atluri S N. On a study of the ~Tc and C*
integrals for fracture analysis under non-steady creep. Engng
Fracture Mechanics , 1982, 16, 625-643.
12. Stonesifer R Band Atluri S N. Moving singularity creep crack
growth analysis with the ~Tc and C* integrals. Engng Fracture
Mechanics, 1982, 769-782.
35
13. Bassani J L and McClintock F A. Creep relaxation of stresses
around a crack tip. IntI Jnl Solids Structures, 1986, 17,
472-492.
14. Brunet M and Boyer J C. A finite element evaluation of path
independent integrals in creeping CT specimens. Numerical Methods
in Fracture Mechanics, 1984. Ed A R Luxmoore and D R J Owen.
Pinewood Press, 1984, Swansea, 519-531.
15. Batte A D, Blackburn W S, Hellen T K and Jackson A D.
Calculation of criteria for the onset of crack propagation in
materials which creep. Numerical Methods in Fracture Mechanics,
1978. Ed A R Luxmoore and D R J Owen. Pinewood Press, Swansea,
487-494.
16. Riedel H. Elastic Plastic Fracture, Vol.1, ASTM STP803. ASTM
STP803, 1983, 505-520.
17. Hui C Y and Riedel H. The asymptotic stress and strain field
near the tip of a growing crack under creep conditions. Intnl Jnl
Fracture, 1981, 17, 409-425.
18. Ainsworth R A. Some observations on creep crack growth. Int Jnl
Fracture, 1982, 20, 147-153.
19. Moran B and Shih C F. Crack tip and associated domain integrals
from momentum and energy balance, Engng Fracture Mechanics, 1987,
29, 615-642.
20. Blackburn W S. Progress report on BERCRAG2 and BERCRAG3, 1987,
CEGB TPRDj0917jR87.
36
MODELLING OF CREEP CRACK GROWTH
G.A. Webster Dept. of Mechanical Engineering Imperial College
London, SW7 2BX
ABSTRACT
Models for describing creep crack growth in terms of linear and
non-linear fracture mechanics concepts are presented. When an
elastic stress field is preserved at a crack tip it is shown that
crack growth rate can be correlated by the stress intensity factor
K and when a creep stress distribution is attained by the creep
fracture parameter C*. However since creep strains of about the
elastic strains only are required for stress redistribution, it is
demonstrated that C* is likely to characterize crack growth in
brittle as well as ductile materials unless creep ductilities as
small as the elastic strains are measured. A procedure for
including ligament damage is included for making residual life
estimates.
INTRODUCTION
In engineering design frequently allowance must be made for the
presence of inherent defects, or for the development of cracks
during service, in assessing the safety of components which are
subjected to severe loading at elevated temperatures. With the
increasing accuracy of non-destructive inspection techniques,
smaller and smaller cracks are being detected and the question of
whether a cracked component can be returned to service, or must be
replaced, is being encountered more often.
The main aim of this paper is to present models of crack growth.
Particular emphasis will be placed on brittle situations. A
fracture mechanics approach will be adopted. The significance of
stress redistribution at the crack tip and damage development in
the ligament ahead of the crack will each be considered. It will be
assumed that creep strain rate e can be described in terms of
stress cr by the uni-axial creep law;
37
n
where Eo' eJo and n are material constants.
CHARACTERIZATIONS OF CREEP CRACK GROWTH
Several parameters have been applied to describe experimental creep
crack growth data. The most commonly used are the stress intensity
factor K, the creep fracture mechanics parameter C* and the
reference stress eJref' Typical relationships for crack propagation
rate a that have been produced are
. a
(2 )
(3)
(4)
where A, H, Do, m, p and <p are material constants which may be
temperature and stress state dependent. Typically it is found that
m = p = nand <P is a fraction close to unity (1,2).
It has been argued that the crack growth process is controlled by
the state of stress and strain rate local to a crack tip. Initially
on loading, in the absence of plasticity, an elastic stress field
is generated ahead of the crack tip as shown in Fig 1. Subsequently
creep deformation will cause stress redistribution. When cracking
commences before stress redistribution, characterizations of crack
growth in terms of K will be expected. For sufficiently high
ductilities and n -7~, crack tip blunting will occur causing the
singularity at the crack tip to be lost, and failure to be
essentially by rupture of the uncracked ligament, making
descriptions in terms of a reference stress more appropriate. For
most values of n, when stLess redistribution is complete the stress
field at the crack tip will be described by C* and correlations of
crack propagation rate as a function of C* will be
anticipated.
Failure by creep rupture is not relevant to relatively brittle
situations. In the next section models of crack growth will be
developed in terms of C* and K to determine the constants in the
experimental relations (eqs (2) and (4» for materials exhibiting a
range of creep properties.
MODELS OF CREEP CRACK GROWTH
The condition when stress redistribution at the crack tip is
complete will be considered first. For this situation, with a creep
law of the form of eq (1), the stress eJij and strain rate Eij
tensors at coordinates (r,e) ahead of tlie crack tip are given by
(3)
38
r'
Figure 1. Elastic and creep stress distributions at a crack tip
together with zone of damage accumulation
I~*er cri·(S,n) [ ]
and
1/ (n+l)
n 0 0 J ( 6)
..- where crij(S,n) and Eij(S,n) are non-dimensional functions of n
and S w1.th In chosen (4) so that their maximum effective
magnitudes are unity. When n = 1, C* predicts the same stress
distribution ahead of a crack tip as K. For n > 1 a stress
distribution similar to that shown in Fig 1 is obtained.
It is possible to use eqs (5) and (6) to develop a model of the
cracking mechanism (5,6). A process zone is postulated at the crack
tip as indicated in Fig 1. It is supposed that this zone
encompasses the region over which damage accumulates locally at the
crack tip. As cracking proceeds, an element of material will first
experience damage when it enters the creep zone at r = rc and will
have accumulated creep strain tij by the time it is a distance r
from the crack tip such that
r
39
(7)
substituting eq (6) into this expression and integrating for a
constant crack velocity a (= -r) assuming that failure occurs when
the creep ductility appropriate to the state of stress at the crack
tip E*f is exhausted, gives
(8)
when the normalizing factor f ij (6,n) is taken as unity.
This expression is relevant to situations where secondary creep
dominates and creep failure strain is constant. It is consistent
with the experimental relation, eq (4), if cp = n/(n+1). For most
materials n > 1 so that rc will be raised to a small fractional
power in eq (8) and relative insensitivity to the process zone size
is predicted. A guide to what value to choose for rc can be gained
from microstructural observations (5,6). Typically these indicate
voiding and microcracking up to several grains ahead of the main
crack tip and it is usually satisfactory to select rc as the
material grain size.
The available ductility E*f at the crack tip is sensitive to the
state of stress there (6). For plane stress conditions it can be
taken to equal the uni-axial creep ductility Ef and for plane
strain situations Ef/50.
The above approach can be extended to materials which undergo
primary, secondary and tertiary creep and which have a decreasing
creep ductility with decrease in stress. A typical creep curve and
stress rupture plot for such a material are shown in Fig 2. An
average creep strain rate EA can be defined in terms of the failure
strain Ef and rupture life tr as
which can be incorporated into eq (1). Similarly for the stress
rupture plot illustrated in Fig 2b),
u
Time t (h)
Figure 2. (a) Simplification of primary, secondary and tertiary
creep data and (b) typical stress rupture plot
where Efa is the uni-axial creep failure strain at stress 00. For n
> U, creep ductility decreases with decrease in stress and for n
= U a constant failure strain is obtained.
For variable creep ductility a constant failure strain cannot be
used in eq (7) as a fracture criterion. Several cumulative damage
models are possible, but when the creep curve is approximated by an
average creep rate fA they can all be reduced to the life fraction
rule (6) which allows the fraction of damage roincurred up to a
given time to be expressed as
ro = J ~t (11) r
and fracture to occur when ro 1 at the crack tip. Consequently
failure takes place at the crack tip when
r c
r o
1 (12)
Substituting eq (10) and then eq (5) into eq (12) and integrating
for a constant crack growth rate gives
U/(n+l)
(n+l-U) /n+l) (13 )
41
For a constant failure strain n = U and eq (13) reduces to eq (8).
It also has the same form as the experimental relation, eq (4), if
u/(n + 1) = •. Equations (8) and (13) can be used to determine the
material constants Do and. in the crack growth law from uni-axial
creep data. Examination of the creep properties of many materials
(6) has indicated that eq (4) can be approximated by
(14)
when a is in rom/h, ef* is a fraction and C* is in MJ/m2h. The
application of eqs (8), (13) and (14) to materials having a wide
range of creep ductilities is shown in Figs 3 to 5. The results
were obtained on specimens of gross thickness B and net thickness
Bn between side-grooves. Figure 3 illustrates the behaviour of a
low alloy steel with a uni-axial creep failure strain of 0.45, Fig
4 the response of a nickel base super-alloy having a ductility of
about 0.15 and Fig 5 that of an aluminium alloy with a failure
strain between 0.07 and 0.02. It is apparent that there is a
progressive trend of correspondence with plane stress predictions
for high ductilities towards agreement with plane strain estimates
with decrease in failure strain.
B
C*{MJ/m2 h)
Figure 3. Creep crack growth data for 21/4 CrMo steel at
538°C
) lVJB
IE
,0
42
Figure 4. Experimental creep crack growth data for nickel-base
superalloy API at 700°C for 0 B = 11 mm, 0 B = 25 mm.
Chained lines are predictions from eq (13)
t-\~ .,:.~
- - Equation (13) 0
~D
C'(MJ/m2 h)
Figure 5. Creep crack growth data for aluminium alloy RR58 at
150°C
43
The plane stress and plane strain bounds of eq (14) are shown
plotted in Fig 6 with aEf as ordinate to produce a material
independent creep crack growth assessment diagram (7). The shaded
area represents the spread of all the e~perimental data found. It
can be seen that the two bounds approximately span the
results.
It has been assumed so far that crack growth does not take place
until after complete stress redistribution at the crack tip. If
crack growth takes place prior to any stress redistribution on
elastic stress field will be preserved and
K --. f(9) J 21tr'
(15)
An elastic stress distribution will always be preserved for a
material which has a creep stress dependence n = 1. For this
situation, sUbstitution of eqs (1) and (15) into eq (7) and
integrating at constant crack growth rate for f (9) = 1 gives
,...... .c -e e 10-......
44
* j;; r
c EfGo
( 16)
for a material with a constant creep ductility. This expression
predicts proportionality between crack growth rate and K and an
increase in crack speed with increase in process zone size. An
equivalent relation can be obtained in terms of C"" by substituting
n = 1 in eq (8). No experimental crack growth data have been
located for a material with n = 1 against which eq (16) can be
compared.
An elastic stress field will also be maintained for a material with
n > 1 if cracking is accompanied by insufficient creep
deformation to allow stress redistribution. When this occurs
substitution of eq (15) into eq (7) for a constant ductility
material with fee) = 1 gives
r
n
-n/2 r dr (17)
(An analysis for a variable ductility material will not be pursued
here since it leads to the same conclusions as when a constant
failure strain is assumed) .
Integration of eq (17) to r = 0 results in an infinite crack growth
rate and a different approach to damage accumulation is required. A
finite crack growth rate can be obtained by postulating that damage
initiates a long way ahead of the crack tip and fracture occurs
when the creep ductility Ef* is exhausted at a distance rc from the
crack tip. With this interpretation eq (17) becomes modified
to
n
Ef r dr (18)
n
Ef r c
( 19)
which corresponds with the experimental relation, eq (2). Unlike
the previous models rc will in general be raised to a power greater
than one and appreciable sensitivity to rc will be expected. It
must be remembered however that rc has a different definition in
the two approaches.
45
SIGNIFICANCE OF REDISTRIBUTION
An indication of the amount of creep strain needed ahead of a crack
tip to achieve stress redistribution can be obtained by making
reference to Fig 1 and eq (5). Generally it is found that the
distance r' ahead of the crack tip at which the elastic and creep
stress distributions intersect is relatively insensitive to the
value of n and close to predictions obtained from limit analysis
methods (8,9). An example for a compact tension specimen of width W
is shown in Fig 7.
0.25,..----------------------____ -,
0.20
O. \('
0.0:.
• n-3 _noS o rr-7 )( rr-l0 o rr-13 .. rr-16 A rr-2O + Haigh and
Richards (9)
O.OO+-----------~----------~------------r_--------~ 0 .2 0.4 0 .6
a/W 0.6 1.0
~igure 7. Relationship between ryW and a/W for a compact tension
test-piece
For most test-piece dimensions r' is at least an order of magnitude
greater than the size of the process zone at a crack tip in which
creep damage is seen to accumulate. This implies that the damage
develops well within the region where stress relaxation takes place
during the redistribution period.
It can be shown, by assuming the stress remains constant at r = r'
during stress redistribution, that the time t' for the creep strain
to equal the elastic strain at this position is given by
t' I
n G
46
where G is the elastic strain energy release rate. This expression
can be compared with one proposed by Riedel and Rice (3) .for the
time to achieve steady state creep conditions
G (21 ) t = ----- 1 (n + l)C*
For a typical value of n ~ 6, t'~ 5tl indicating that stress
redistribution is expected to be substantially complete when a
creep strain of about one fifth of the corresponding elastic value
is accumulated at r = r'. From eq (