1
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
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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
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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 (