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Calculation of critical stress intensity factors of nuclear graphite from small specimen tests
ooO8-6223185 $3.00+ .oO 0 1985 Pergamon Press Ltd.
Carbon Vol. 23. No. 4. pp 3X7-393. 1985 Pnnted ,n the U.S A
CALCULATION OF CRITICAL STRESS INTENSITY FACTORS OF NUCLEAR GRAPHITE FROM SMALL SPECIMEN TESTS
Central Electricity Generating
A. P. G. ROSE Board, Berkeley Nuclear Laboratories. CL13 9PB. U.K.
Berkeley,
Glos
(Received 30 July 1984) Abstract-Measurements of failure loads, deflections and acoustic emission response of different sized small beams of virgin pitchcoke graphite are reported. Methods of analysing the data to obtain the critical stress intensity factor K,,, are considered. Values determined as a function of failure load and critical crack size are found to agree with those from tests on whole components. In principle it is feasible to apply this method to small samples of irradiated
graphite to determine the variation of fracture toughness
1. INTRODUCTION In gas cooled nuclear reactors graphite components are routinely sampled to confirm expected performance. There is provision within the UK graphite monitoring programme for determination of bend strength and dynamic modulus with increasing reactor dwell. However, no measurements of the variation of critical stress intensity factor K,,. during life are currently made. Such information is desirable due to the wide application of fracture mechanics theories. A statistical model of graphite failure[ l] capable of predicting failure stresses of small undefected beams in bend and tension has K,, as its highest ranked input variable. It is envisaged that this fracture model will be employed in conjunction with finite element stress analysis to calculate failure probabilities of in-service components and in this case a knowledge of the variation of K,, with irradiation is essential. There are however considerable difficulties in performing fracture toughness tests on graphite, both in production of a suitable test piece and interpretation of the results. Standard methods of K,,measurement[2,3] strictly only apply to metallic specimens containing a sharp crack fatigued into the specimen beneath a machined defect. Component shape frequently precludes obtaining specimens of standard dimensions and crack branching occurs when attempts are made to fatigue crack the inhomogeneous material structure. This paper reports fracture toughness data from non-standard notched beams cut from reactor components and broken in bending. An extensive series of these tests on virgin pitchcoke graphite are analysed and the extension of the technique to irradiated material considered.
with irradiation/burn
up.
defects of 0.5 mm root radius and depth from 0.5 to 7.9 mm. From Fig. 1, type (i) notches had an included angle of 60” and type (ii) vertical sides (0” included angle). No difference in failure loads between these types of notches of the same depth aor was obtained. Measurements of failure deflection for samples at selected values of a,, throughout the range examined were also made. The variation of acoustic emission response with increasing notch
(1)
3-Point
emission amplifiers
Bending of Curved Ears, Width B=12.0mm.
\‘\,
‘\
i)
I
//I I/
\ \
k102.7mm
\\ \\
ii ) 00 Ill
.._
(2)
&Point
Bcndmg of Strarght Bars, Width k
32mm-
B=lO.Omm
cl
2. EXPERIMENTAL h
Details of specimen dimensions, machined notches and bend tests performed are shown in Fig. 1. All tests were such that the direction of tensile stress was perpendicular to the extrusion direction. Thirty curved bar specimens were tested without notches and eighty were notched with
64 mm
-I 60’ A--
Fig.
387
1. Schematic representation
of bend tests performed notched small specimens of pitchcoke graphite.
a0
on
A. P. G. ROSE depth was determined by coupling a small piezoelectric transducer to the beam and analysing the resulting signals with Endevco 3000 series acoustic emission amplifiers set in the amplitude mode. Twenty five straight bars were tested intact and sixty were notched with type (i) notches of again 0.5 mm root radius and depth from 0.1 to 3.0 mm. For an elastic test the maximum tensile stress in bending in a curved bar is 5% higher than that in a straight beam of the same thickness and width under identical loading. Where required therefore the failure stresses of curved beams have been corrected to those of equivalent straight beams. A small amount of quantitative image analysis of the large pore structure was carried out. The contrast between pores (dark) and material (light) was enhanced by evaporating a thin layer of gold onto a polished graphite surface cut parallel to the extrusion axis. The maximum chord lengths of all pores above a minimum cut off value of 0.3 mm were determined. 3, METHODS OF ASSESSING FRACTURE TOUGHNESS
3.1 Po~y~omiaiExpressions The stress intensity factor K,, characterising the elastic stresses around a sharp crack of length a, under tensile loading may be written K, = Yuat
(1)
where Yis a geometrical factor and u is the applied stress. Polynomial expressions exist for the evaluation of Yunder different loading conditions{41 as a function of the aspect ratio a/W of a crack in a specimen of thickness W. In the critical case therefore K,cvalues may be obtained from the nomial failure stress and the crack depth. For expressions of this type to be of use in graphite two assumptions regarding this crack depth are necessary. Firstly, that there is an effective additional crack below the machined notch due to the flaw dis~bution present. This is discussed fully in Section 4 where it is deduced that consistent values of K,, from bend specimens containing notches of various depths are obtained by assuming a through thickness additional crack length of 0.6 mm. Secondly, as there is a distribution of pores already present it is impossible to obtain the normal fatigue sharpened crack used in metallic specimens and therefore a machined defect of length a, mm and root radius -0.5 mm with a crack of length 0.6 mm at its root is taken to be equivalent to a sharp crack of overall length (a, f 0.6) mm. The validity of this assumption has been tested using the BERSAFE finite element codef51 and good agreement of calculated stress intensity factors from these two cases was obtained. For a beam in pure bending confining a sharp crack of length a, Srawley and Cross[4] give the stress intensity factor K, as K, = o,,, w [ 1.9887 - 1.326~ - (3.49 - 0.68X + 1,35x*)x(1 -x)/(1
+x)*1
with earlier versions[6]. Modified forms exist to account for the shearing forces present in three-point bend at specific span to specimen depth ratios. However these predict K, values within 2% of the pure bending case and therefore for ease of calculation eqn (2) has been used for the bend tests reported. 3.2 Critical Crack E~te~~*~ Force A second method of calculating the critical stress intensity factor is to evaluate the critical crack extension force, G,, per unit length of crack front (equal to the rate of change of energy release with crack extension). In an elastic test the load displacement record may be used to evaluate this quantity. Consider Fig. 2 which shows a schematic load displacement plot of two elastic lines for samples containing notches of length (a,) and (a, + da,) respectively. The gentler slope of the (a, + da,) line reflects the lower effective modulus at an increased notch depth. The specimen notched to length (a,) fails at load P and deflection 6, and the (a, + da,) notched specimen fails at a lower load and deflection, point E. If the specimen (a,) is loaded to point A then it contains a greater stored energy than specimen (a, + da,) at the same deflection, point E, by an amount equal to the area shaded, triangle OAE. Hence going from notch size (a,) to size (a, + da,) the release of energy that provides the crack extension force is given by this area such that for a small increment of crack growth da,: Energy release = Area triangle OAE = triangle OK - triangle OED - rectangle ABCD = SdPJ2 - PdSi2
_____--_-s--------1
(2)
where uM is the maximum nominal stress at the root of a crack of aspect ratio x. This expression is compatible
(3)
Deflection
Fig. 2. Schematicassentation specimens containing
-
of load deflection curves for notches of depth a, and a, + da, respectively.
389
Calculation of critical stress intensity factors
where dp and d8 are the associated decreases in failure load and extension. The rate of change of energy release with crack extension in a specimen of width B is then G,, = (SdPlda,
- Pd8ldaJl2B.
(4)
Thus knowing failure loads and deflections and their variation with notch depth the critical crack extension force may be calculated. In plane stress G,, is simply equal to K,,‘IE where E is Young’s modulus. Under plane strain conditions the modulus is replaced by E/( 1 - uzf although the low value of Poisson’s ratio u = 0.17, normally assumed for graphite means that this correction may be neglected. 4. RESULTS AND
DISCUSSION
In comparing results from specimens of different geometry it is useful to nondimensiona~ise the load and deflection data as a function of notch depth so as to obtain one normalised curve in each case. This may be most conveniently achieved in terms of the failure load P,, and failure deflection S,, of identical unnotched beams. Then the nondimensional failure loads and deflections may be defined as P’ = PIP, and 6’ = S/6,, respectively. Figure 3 shows the variation of P’ and 6’ with increasing notch aspect ratio. It is seen from the figure that failure deflections remain constant at -0.9 of the unnotched value independent of notch aspect ratio over the range examined. From this data it is possible to a first approximation to simpli~ eqn (4) by taking d&/da, = 0 and nondimensionaiising dP/da, such that
G,, = 6P,, (dP’ldx’)/2BW
Ouantity :oilure Dellaction Fai Lure Load
(5)
Notch fnctuded Angle
wherex’ is the ratio of notch depth to specimen thickness. This simplified form may be readily used to evaluate G,, If additional deflection measurements were available it would be possible to include a nonzero dimensionless term d8’ldx’ although the shallow slope suggested means that this would be small in comparison with the dP’ldx’ term. Thus eqn (5) has been used to cafcuiate the criticaf crack extension force in the present paper. Consider further Fig. 3. At a,lW values of 0.01 and 0.02 not all samples failed from the notch. The insert to the figure indicates that certain failure from the notch root occurs only for notch depths >0.3 mm in a 10 mm thick bar. This value must be regarded as app~ximate as it is based on a limited number of results. However, it does give a measure of the largest cracks present within the material at fracture and indicates the dividing line between one of these cracks and the machined notch being the more severe defect. From approximately a,,/ W > 0.03 the specific loads P’, fall off along a smooth curve. Error bars shown represent variations of * one standard deviation in both P and P,,. Results from straight and curved beams containing notches of the same root radius tested in three- and four-point bend all lie on the same experimental curve and thus allow (dP’/dx’) required in eqn (5) to be obtained. The agreement of the notched beam results with this single curve is very good. For x > 0.52 the P’ curve is extrapolated to zero failure load at a through thickness notch. The failure deflection measurements corrected for the testing machine deflection, are also plotted in a dimensionless form in Fig. 3. The error bars include i one standard deviation in both 6 and 6,,. Unlike the Ioad results, as mentioned above the deflections remain constant at approximately 0.9 of the unnotched value inde-
Band
*bo’
B4am Type
A
Curved
0’
3-Point
x
Curved
60’
3 - Point
0
Curved
0’
3-Point
0
Straight
60’
f-Point
1-t
I
D$th
I
of %ch
( mm )
0.8
0.6
Notch depth I specimen
0.,,
y--_-, 0.0 thickness,
0.9 -
1.0
x’-
Fig. 3. The variation of normalised failure loads and deflections with aspect ratio of notch-extrapolated X’ = 0.52.
beyond
390
A. P. G. ROSE
pendent of notch aspect ratio. There is a slightly larger amount of scatter in the deflection measurements and no correlation between load and deflection at failure was found. It appears to be a reasonable assumption as outlined above that failure deflection is independent of notch depth within the range examined. In using eqn (5) to calculate G,,, account must be taken of the effect of root radius of the notch on the term (dP’/ dx’). Brocklehurst[7] has reported four-point bend tests on beams of find grained Gilsocarbon graphite containing notches of different aspect ratios and root radii from -0.003 mm to 10.2 mm. These results are illustrated in Fig. 4. Not all the standard deviations on failure loads are given in the source paper therefore these are omitted but appear from the original to be of the same order as those of Fig. 3. Also shown on Fig. 4 are the P’ curve from Fig. 3 and the expected decrease in normalised failure load for an unnotched beam as its thickness is reduced. This is obtained from the elastic bend stress formula such that P' is proportional to thickness squared assuming that failure occurs at a constant maximum ligament stress. The elastic line falls more gently than the P' curve over the range of aspect ratios examined indicating the greater stress intensification at a notch root compared with a flat beam of thickness reduced by the notch depth. The finest root radii of -0.003 mm were obtained by tapping a coarser notch root with a razor blade. It is seen in Fig. 4 that this produces little further stress intensification than the 0.5 mm root radii notches in the pitchcoke graphite. This is in agreement with observations made by Brocklehurst[7] in that the material effectively blunts any sharper notches to a limiting value of root radius. Hence, the assumption of the 0.5 mm machined notches being ‘sharp’ is further justified. Considering the
Unnotched
Notch
results at x’ = 0.2, for which most data is available, the stress intensification is seen to reduce as the notch becomes blunter until at the largest radius of 10.2 mm the beam is equivalent to an un-notched elastic specimen of the same thickness as the ligament length below the notch root. Results for notches of other aspect ratios are also well enveloped by the P' curve at the ‘sharp’ extreme and the elastic curve for unnotched beams. It has been previously mentioned that in order to use a polynomial function such as eqn (2) to calculate consistent K,,values in graphite it must be assumed that the machined notch is equivalent to a sharp crack and that there is an additional crack below the notch root due to the distribution of pores. The first of these assumptions has been justified with a BERSAFE comparison and use of the Brocklehurst data. We need now to discuss further the second assumption. The insert to Fig. 3 seeks to determine the size of this additional crack by defining the minimum depth at which the artificial notch acts certainly as the fracture initiation site. This approach was first adopted by Brocklehurst and Kelly[8]. It assumes that the porosity distribution may be conveniently represented as a single through thickness crack for the purpose of K,,measurement, although this may be criticised in that the additional crack size found by Brocklehurst and Kelly is different in bend and tension and therefore is a function of stress distribution. However, as we are presently concerned simply with bend tests the idea is of considerable value. An additional crack size of 0.3 mm is indicated from Fig. 3. This must be treated with some caution as its choice is mainly based on 6 tests on bars containing 0.4 mm notches. If more of these tests were done then a probability of failure from the notch of less than unity may well be obtained. Therefore, the value of 0.3 mm must be regarded as a lower limit estimate.
beam
depth I specimen
thickness,
Fig. 4. Data[7] on the effect of notch root radius on failure loads of Gilsocarbon various aspect ratios.
x’ beams containing
notches of
Calculation
of critical stress intensity factors
creases. A selection of the notched beam results are shown as broken lines on the figure. For these, the dependence of K,, on additional crack length is reduced as the additional crack becomes a smaller proportion of the total crack length. The interesting feature of the figure is the approximate cross-over point of the notched and unnotched cases at which the K,, value given by eqn (2) is independent on machined notch depth. This occurs at an additional crack length of approximately 0.6 mm. The independence of the calculated K,, value to notch depth indicates that notched or unnotched specimens may be used in fracture toughness testing by this method. The second approach to calculating the critical stress intensity factor requires use of load and deflection measurements at failure to give the critical crack extension force per unit area of crack from eqn (5). Results are shown in Fig. 6 and it is seen that the K,, value is calculated to decrease as the notch depth increases. Physically, this means that the rate of energy release with crack extension is a function of the initial notch depth. There is support for this in the acoustic emissibn data summarised in Table 1. Noise detected prior to failure has been associated with microcracking and hence energy release in addition to that required to form the main failure crackj IO]. For the 30 unnotched curved beams tested in three-point bending as in Fig. 1, the onset of noise was
A second piece of evidence is available from image analysis studies. Using the technique explained in Section 2, a polished surface parallel to extrusion so containing the most probable fracture initiation sites was examined. All pores having a maximum chord length >0.3 mm were analysed. The largest chord length obtained was 1.6 mm and the average was 0.5 + 0.2 mm. This average cannot be regarded as absolute as it is a function of the lower limit cut off value. In addition, the orientation of large pores is typically perpendicular to the crack growth direction and these pores would thus be relatively ineffectual in acting as cracks. The image analysis of large pores is therefore inconclusive in defining an additional crack depth. Rather it is more likely the case that the additional crack consists of an agglomeration of much smaller pores along the notch root which coalese during the test thus acting as a single crack immediately prior to failure. The polynomial expression of eqn (2) may be further utilised to study the additional crack length. Figure 5 shows how additions to the machined notch depth affect the fracture toughness predicted from this equation. A figure of this type has been deduced by Birch et a/.[91 for a wide range of different graphites. The full line represents the case of an unnotched beam, the K,, value rising rapidly as the assumed additional crack size in-
0.2
0.4
Additional
0.6
0.6
crack
391
1.0
1.2
1.4
1.6
length (mm) -
Fig. 5. K,, values given by the Srawley and Gross polynominal expression as a function of the additional crack length added to the machined defect.
392
A. P. G. ROSE
Symbol Bcam tYP
t.6 t
Notch included K ,c calcutation angle.
x
Curved
60.
SlwleYardoror
l
CUNcd
0.
kawkyandQross
Q
Straight
60.
0
UNed
0.
Srawtqand Bror Critical crack
crtcnsian
force
0.6 P
0
0.1
0.2
Notch
depth
0.4 0.3 I specimen
0.5
0.6
thickness,
x’ -
Fig. 6. The variation of critical stress intensity factors calculated by different methods as a function of notch depth/specimen thickness ratio.
detected at a load of 130 N 2 19 N. This is 15% t 4% of the final failure load and indicates widespread secondary cracking during the test. With a specimen contaming a notch, acoustic emission is detected at a lower load which tends to decrease as the notch deepens, although the results are widely scattered. When the fraction of onset load and failure load is considered, it is evident that cracking starts later in the test as the notch length increases. If this trend is maintained for deeper notches then no cracking will be detected before failure and all the energy release with crack extension during the test will go towards creating the primary crack. To determine the point at which this occurs and hence a lower limit
fracture toughness value requires tests on specimens containing notches of aspect ratio greater than 0.5. Equation (5) predicts that a lower limit G,, is obtained at the minimum value of dp’ldr’. This occurs as x’ + 1 that is as the notch depth approaches the specimen thickness. However, such tests are not easy to interpret due to the proximity of the surface towards which the growing crack propagates. Hence there are considerable experimental difficulties in defining a test to measure a consistent value of critical stress intensity factor using the method of energy release with crack extension. Figure 6 is a plot of critical stress intensity factor K,<, against notch aspect ratio for all small specimen data as
Table 1. Acoustic emission response of notched graphite beams in three-point bend Aspect Ratio of Notch
Load at Onset of Acoustic Emission + 1 Standard Deviation
0
(N)
Fraction of Failure Load at Onset of Acoustic Emission f 1 Standard Deviation
130
f
19
0.15
* 0.04
0.25
80
f
12
0.25
i
0.05
0.36
69
+ 19
0.31
f
0.11
0.48
69
f
0.42
*
0.27
32
393
Calculation of critical stress intensity factors calculated from the two methods described. Following from the discussion above the values obtained from the critical crack extension force are seen to be a decreasing function of notch aspect ratio. Values calculated from the Srawley and Gross polynomial, eqn (2), applied to the 10 mm thick straight beams are also seen to decrease sharply at x’ 2 0.2. In these cases there is 8 mm or less thickness below the notch and the decrease in K,, is associated with a grain size effect[7] as the reduced section is untypical of bulk behaviour. However, all the data from curved beams contains a thickness greater than 8 mm beneath the notch and therefore does not show this effect. The independence to notch depth of K,, values obtained from the polynomial expression is noted, the spread of results over +- one standard deviation being approximately 1.2 +- 0.15 MNm-“I in good agreement with data from whole components[ 11). It appears that the choice of an additional crack length of 0.6 mm below the machined notch compensates well for the decrease in critical crack extension force with increasing notch depth throughout the range examined.
5.
FEASIBILITY
DETERMINATION
OF FRACTURE TOUGHNESS FROM SMALL SPECIMENS OF
IRRADIATED
NUCLEAR GRAPHITE
From the above discussion, the method of fracture toughness determination based on critical crack extension force has been shown to be complex in its interpretation. A further disadvantage is that it requires load and deflection measurements at failure and assumes that deflection at failure is constant independent of notch length. The most promising analysis is clearly the Srawley and Gross approach. This only requires a knowledge of the failure load and additional crack length. For virgin graphite a good agreement between small specimen and bulk K,, values has been obtained by assuming that the material contains an additional crack length of 0.6 mm. However, the question remains of the variation of this additional crack length with irradiation. From the limited amount of image analysis performed it appears that it is not simply controlled by the largest pores present but rather is a more complicated function of the total porosity distribution. Radiolytic oxidation only takes place in the open pores. Hence, image analysis or mercury porosimetry comparing the porosity distributions in virgin and irradiated material are possible techniques to understand any change in the additional crack length with irradiation. Acoustic emission from bend tests will also indicate the onset and extent of microcracking in the fracture of irradiated material which will have a direct effect on the additional crack length. The most immediate method of determining its variation is to perform a series of bend tests using acoustic emission on specimens cut from irradiated components at a range of bum-ups, some samples from each component containing notches of known depth and 0.5 mm root radius. By comparing the fracture loads of notched and unnotched specimens from the same component, a curve to give the additional crack length in the manner of Fig. 5 may be deduced and hence the dependence of the additional crack length on bum-up obtained. In this
manner a consistent method of measuring the variation of K,, with irradiation will be established. Any further work to investigate changes in porosity distribution with bum-up may then be considered. 6. CONCLUSIONS
Measurements of failure loads, deflections and acoustic emission response from notched and unnotched small beams of virgin pitchcoke graphite are reported. Methods of calculating the critical stress intensity factor K,, , are presented and the feasibility of defining a test to obtain K,, values from similar irradiated specimens is discussed. The following conclusions have been drawn. 6.1 A technique based on measurement of the critical crack extension force is complex in its interpretation, requires data on load and deflection at failure and is not considered worthy of further pursuit. 6.2 A consistent method of determining K,, from a polynomial function of failure load and critical crack size has been justified for small beams of a pitchcoke graphite either with or without notches. Values obtained are in agreement with those from tests on whole components. 6.3 In order to determine K,, as a function of irradiation this method requires knowledge of the variation with irradiation of an ‘additional crack length’ below any machined notch. This will be obtained by cutting small specimens from components at a series of known burnups, machining notches of known length into a proportion of the samples from each and testing all of them in threepoint bend. Acknowledgement-This paper is published by permission of the Central Electricity Generating Board. REFERENCES 1.
2. 3. 4. 5. 6. 7.
8. 9. 10. Il.
A. P. G. Rose, L. S. Field and M. 0. Tucker, Exrended Absrracts 16th Biennial Cons. on Carbon, San Diego, 402 (1983). ASTM, E399-78 (1978). British Standards Institute, BS5447 (1977). J. E. Srawley and 9. Gross, ASTM Special Technical Publication 60 I ( 1976). M. 9. Watson, CEGB, Private communication (1983). W. F. Brown and J. E. Srawley, ASTM Special Technical Publication 410,(1966). J. E. Brocklehurst, In Chemistry and Physics of Carbon. (Edited by P. L. Walker, Jr.), 13, 145-283, Marcel Dekker, New York (1974). J. E. Brocklehurst and 9. T. Kelly, Proc. fAEA Conf. Mechanical Behavior of Graphite for HTRs’,Gif-sur-Yvette. IAEA Vienna, 42-50 (1980). M. Birch, R. G. Brown and J. E. Brocklehurst, Exrended Abstracts 16th Biennial Conf. on Carbon, San Diego, 404 (1983). I. M. Pickup, R. G. Cooke and 9. McEnaney, Exrended Abstracts 15th Biennial Conf. on Carbon, Philadelphia, 506 (1981). A. P. G. Rose and M. 0. Tucker, J. of Nuclear Materials, 110, 186-195 (1982).
Carbon Vol. 23. No. 4. pp 3X7-393. 1985 Pnnted ,n the U.S A
CALCULATION OF CRITICAL STRESS INTENSITY FACTORS OF NUCLEAR GRAPHITE FROM SMALL SPECIMEN TESTS
Central Electricity Generating
A. P. G. ROSE Board, Berkeley Nuclear Laboratories. CL13 9PB. U.K.
Berkeley,
Glos
(Received 30 July 1984) Abstract-Measurements of failure loads, deflections and acoustic emission response of different sized small beams of virgin pitchcoke graphite are reported. Methods of analysing the data to obtain the critical stress intensity factor K,,, are considered. Values determined as a function of failure load and critical crack size are found to agree with those from tests on whole components. In principle it is feasible to apply this method to small samples of irradiated
graphite to determine the variation of fracture toughness
1. INTRODUCTION In gas cooled nuclear reactors graphite components are routinely sampled to confirm expected performance. There is provision within the UK graphite monitoring programme for determination of bend strength and dynamic modulus with increasing reactor dwell. However, no measurements of the variation of critical stress intensity factor K,,. during life are currently made. Such information is desirable due to the wide application of fracture mechanics theories. A statistical model of graphite failure[ l] capable of predicting failure stresses of small undefected beams in bend and tension has K,, as its highest ranked input variable. It is envisaged that this fracture model will be employed in conjunction with finite element stress analysis to calculate failure probabilities of in-service components and in this case a knowledge of the variation of K,, with irradiation is essential. There are however considerable difficulties in performing fracture toughness tests on graphite, both in production of a suitable test piece and interpretation of the results. Standard methods of K,,measurement[2,3] strictly only apply to metallic specimens containing a sharp crack fatigued into the specimen beneath a machined defect. Component shape frequently precludes obtaining specimens of standard dimensions and crack branching occurs when attempts are made to fatigue crack the inhomogeneous material structure. This paper reports fracture toughness data from non-standard notched beams cut from reactor components and broken in bending. An extensive series of these tests on virgin pitchcoke graphite are analysed and the extension of the technique to irradiated material considered.
with irradiation/burn
up.
defects of 0.5 mm root radius and depth from 0.5 to 7.9 mm. From Fig. 1, type (i) notches had an included angle of 60” and type (ii) vertical sides (0” included angle). No difference in failure loads between these types of notches of the same depth aor was obtained. Measurements of failure deflection for samples at selected values of a,, throughout the range examined were also made. The variation of acoustic emission response with increasing notch
(1)
3-Point
emission amplifiers
Bending of Curved Ears, Width B=12.0mm.
\‘\,
‘\
i)
I
//I I/
\ \
k102.7mm
\\ \\
ii ) 00 Ill
.._
(2)
&Point
Bcndmg of Strarght Bars, Width k
32mm-
B=lO.Omm
cl
2. EXPERIMENTAL h
Details of specimen dimensions, machined notches and bend tests performed are shown in Fig. 1. All tests were such that the direction of tensile stress was perpendicular to the extrusion direction. Thirty curved bar specimens were tested without notches and eighty were notched with
64 mm
-I 60’ A--
Fig.
387
1. Schematic representation
of bend tests performed notched small specimens of pitchcoke graphite.
a0
on
A. P. G. ROSE depth was determined by coupling a small piezoelectric transducer to the beam and analysing the resulting signals with Endevco 3000 series acoustic emission amplifiers set in the amplitude mode. Twenty five straight bars were tested intact and sixty were notched with type (i) notches of again 0.5 mm root radius and depth from 0.1 to 3.0 mm. For an elastic test the maximum tensile stress in bending in a curved bar is 5% higher than that in a straight beam of the same thickness and width under identical loading. Where required therefore the failure stresses of curved beams have been corrected to those of equivalent straight beams. A small amount of quantitative image analysis of the large pore structure was carried out. The contrast between pores (dark) and material (light) was enhanced by evaporating a thin layer of gold onto a polished graphite surface cut parallel to the extrusion axis. The maximum chord lengths of all pores above a minimum cut off value of 0.3 mm were determined. 3, METHODS OF ASSESSING FRACTURE TOUGHNESS
3.1 Po~y~omiaiExpressions The stress intensity factor K,, characterising the elastic stresses around a sharp crack of length a, under tensile loading may be written K, = Yuat
(1)
where Yis a geometrical factor and u is the applied stress. Polynomial expressions exist for the evaluation of Yunder different loading conditions{41 as a function of the aspect ratio a/W of a crack in a specimen of thickness W. In the critical case therefore K,cvalues may be obtained from the nomial failure stress and the crack depth. For expressions of this type to be of use in graphite two assumptions regarding this crack depth are necessary. Firstly, that there is an effective additional crack below the machined notch due to the flaw dis~bution present. This is discussed fully in Section 4 where it is deduced that consistent values of K,, from bend specimens containing notches of various depths are obtained by assuming a through thickness additional crack length of 0.6 mm. Secondly, as there is a distribution of pores already present it is impossible to obtain the normal fatigue sharpened crack used in metallic specimens and therefore a machined defect of length a, mm and root radius -0.5 mm with a crack of length 0.6 mm at its root is taken to be equivalent to a sharp crack of overall length (a, f 0.6) mm. The validity of this assumption has been tested using the BERSAFE finite element codef51 and good agreement of calculated stress intensity factors from these two cases was obtained. For a beam in pure bending confining a sharp crack of length a, Srawley and Cross[4] give the stress intensity factor K, as K, = o,,, w [ 1.9887 - 1.326~ - (3.49 - 0.68X + 1,35x*)x(1 -x)/(1
+x)*1
with earlier versions[6]. Modified forms exist to account for the shearing forces present in three-point bend at specific span to specimen depth ratios. However these predict K, values within 2% of the pure bending case and therefore for ease of calculation eqn (2) has been used for the bend tests reported. 3.2 Critical Crack E~te~~*~ Force A second method of calculating the critical stress intensity factor is to evaluate the critical crack extension force, G,, per unit length of crack front (equal to the rate of change of energy release with crack extension). In an elastic test the load displacement record may be used to evaluate this quantity. Consider Fig. 2 which shows a schematic load displacement plot of two elastic lines for samples containing notches of length (a,) and (a, + da,) respectively. The gentler slope of the (a, + da,) line reflects the lower effective modulus at an increased notch depth. The specimen notched to length (a,) fails at load P and deflection 6, and the (a, + da,) notched specimen fails at a lower load and deflection, point E. If the specimen (a,) is loaded to point A then it contains a greater stored energy than specimen (a, + da,) at the same deflection, point E, by an amount equal to the area shaded, triangle OAE. Hence going from notch size (a,) to size (a, + da,) the release of energy that provides the crack extension force is given by this area such that for a small increment of crack growth da,: Energy release = Area triangle OAE = triangle OK - triangle OED - rectangle ABCD = SdPJ2 - PdSi2
_____--_-s--------1
(2)
where uM is the maximum nominal stress at the root of a crack of aspect ratio x. This expression is compatible
(3)
Deflection
Fig. 2. Schematicassentation specimens containing
-
of load deflection curves for notches of depth a, and a, + da, respectively.
389
Calculation of critical stress intensity factors
where dp and d8 are the associated decreases in failure load and extension. The rate of change of energy release with crack extension in a specimen of width B is then G,, = (SdPlda,
- Pd8ldaJl2B.
(4)
Thus knowing failure loads and deflections and their variation with notch depth the critical crack extension force may be calculated. In plane stress G,, is simply equal to K,,‘IE where E is Young’s modulus. Under plane strain conditions the modulus is replaced by E/( 1 - uzf although the low value of Poisson’s ratio u = 0.17, normally assumed for graphite means that this correction may be neglected. 4. RESULTS AND
DISCUSSION
In comparing results from specimens of different geometry it is useful to nondimensiona~ise the load and deflection data as a function of notch depth so as to obtain one normalised curve in each case. This may be most conveniently achieved in terms of the failure load P,, and failure deflection S,, of identical unnotched beams. Then the nondimensional failure loads and deflections may be defined as P’ = PIP, and 6’ = S/6,, respectively. Figure 3 shows the variation of P’ and 6’ with increasing notch aspect ratio. It is seen from the figure that failure deflections remain constant at -0.9 of the unnotched value independent of notch aspect ratio over the range examined. From this data it is possible to a first approximation to simpli~ eqn (4) by taking d&/da, = 0 and nondimensionaiising dP/da, such that
G,, = 6P,, (dP’ldx’)/2BW
Ouantity :oilure Dellaction Fai Lure Load
(5)
Notch fnctuded Angle
wherex’ is the ratio of notch depth to specimen thickness. This simplified form may be readily used to evaluate G,, If additional deflection measurements were available it would be possible to include a nonzero dimensionless term d8’ldx’ although the shallow slope suggested means that this would be small in comparison with the dP’ldx’ term. Thus eqn (5) has been used to cafcuiate the criticaf crack extension force in the present paper. Consider further Fig. 3. At a,lW values of 0.01 and 0.02 not all samples failed from the notch. The insert to the figure indicates that certain failure from the notch root occurs only for notch depths >0.3 mm in a 10 mm thick bar. This value must be regarded as app~ximate as it is based on a limited number of results. However, it does give a measure of the largest cracks present within the material at fracture and indicates the dividing line between one of these cracks and the machined notch being the more severe defect. From approximately a,,/ W > 0.03 the specific loads P’, fall off along a smooth curve. Error bars shown represent variations of * one standard deviation in both P and P,,. Results from straight and curved beams containing notches of the same root radius tested in three- and four-point bend all lie on the same experimental curve and thus allow (dP’/dx’) required in eqn (5) to be obtained. The agreement of the notched beam results with this single curve is very good. For x > 0.52 the P’ curve is extrapolated to zero failure load at a through thickness notch. The failure deflection measurements corrected for the testing machine deflection, are also plotted in a dimensionless form in Fig. 3. The error bars include i one standard deviation in both 6 and 6,,. Unlike the Ioad results, as mentioned above the deflections remain constant at approximately 0.9 of the unnotched value inde-
Band
*bo’
B4am Type
A
Curved
0’
3-Point
x
Curved
60’
3 - Point
0
Curved
0’
3-Point
0
Straight
60’
f-Point
1-t
I
D$th
I
of %ch
( mm )
0.8
0.6
Notch depth I specimen
0.,,
y--_-, 0.0 thickness,
0.9 -
1.0
x’-
Fig. 3. The variation of normalised failure loads and deflections with aspect ratio of notch-extrapolated X’ = 0.52.
beyond
390
A. P. G. ROSE
pendent of notch aspect ratio. There is a slightly larger amount of scatter in the deflection measurements and no correlation between load and deflection at failure was found. It appears to be a reasonable assumption as outlined above that failure deflection is independent of notch depth within the range examined. In using eqn (5) to calculate G,,, account must be taken of the effect of root radius of the notch on the term (dP’/ dx’). Brocklehurst[7] has reported four-point bend tests on beams of find grained Gilsocarbon graphite containing notches of different aspect ratios and root radii from -0.003 mm to 10.2 mm. These results are illustrated in Fig. 4. Not all the standard deviations on failure loads are given in the source paper therefore these are omitted but appear from the original to be of the same order as those of Fig. 3. Also shown on Fig. 4 are the P’ curve from Fig. 3 and the expected decrease in normalised failure load for an unnotched beam as its thickness is reduced. This is obtained from the elastic bend stress formula such that P' is proportional to thickness squared assuming that failure occurs at a constant maximum ligament stress. The elastic line falls more gently than the P' curve over the range of aspect ratios examined indicating the greater stress intensification at a notch root compared with a flat beam of thickness reduced by the notch depth. The finest root radii of -0.003 mm were obtained by tapping a coarser notch root with a razor blade. It is seen in Fig. 4 that this produces little further stress intensification than the 0.5 mm root radii notches in the pitchcoke graphite. This is in agreement with observations made by Brocklehurst[7] in that the material effectively blunts any sharper notches to a limiting value of root radius. Hence, the assumption of the 0.5 mm machined notches being ‘sharp’ is further justified. Considering the
Unnotched
Notch
results at x’ = 0.2, for which most data is available, the stress intensification is seen to reduce as the notch becomes blunter until at the largest radius of 10.2 mm the beam is equivalent to an un-notched elastic specimen of the same thickness as the ligament length below the notch root. Results for notches of other aspect ratios are also well enveloped by the P' curve at the ‘sharp’ extreme and the elastic curve for unnotched beams. It has been previously mentioned that in order to use a polynomial function such as eqn (2) to calculate consistent K,,values in graphite it must be assumed that the machined notch is equivalent to a sharp crack and that there is an additional crack below the notch root due to the distribution of pores. The first of these assumptions has been justified with a BERSAFE comparison and use of the Brocklehurst data. We need now to discuss further the second assumption. The insert to Fig. 3 seeks to determine the size of this additional crack by defining the minimum depth at which the artificial notch acts certainly as the fracture initiation site. This approach was first adopted by Brocklehurst and Kelly[8]. It assumes that the porosity distribution may be conveniently represented as a single through thickness crack for the purpose of K,,measurement, although this may be criticised in that the additional crack size found by Brocklehurst and Kelly is different in bend and tension and therefore is a function of stress distribution. However, as we are presently concerned simply with bend tests the idea is of considerable value. An additional crack size of 0.3 mm is indicated from Fig. 3. This must be treated with some caution as its choice is mainly based on 6 tests on bars containing 0.4 mm notches. If more of these tests were done then a probability of failure from the notch of less than unity may well be obtained. Therefore, the value of 0.3 mm must be regarded as a lower limit estimate.
beam
depth I specimen
thickness,
Fig. 4. Data[7] on the effect of notch root radius on failure loads of Gilsocarbon various aspect ratios.
x’ beams containing
notches of
Calculation
of critical stress intensity factors
creases. A selection of the notched beam results are shown as broken lines on the figure. For these, the dependence of K,, on additional crack length is reduced as the additional crack becomes a smaller proportion of the total crack length. The interesting feature of the figure is the approximate cross-over point of the notched and unnotched cases at which the K,, value given by eqn (2) is independent on machined notch depth. This occurs at an additional crack length of approximately 0.6 mm. The independence of the calculated K,, value to notch depth indicates that notched or unnotched specimens may be used in fracture toughness testing by this method. The second approach to calculating the critical stress intensity factor requires use of load and deflection measurements at failure to give the critical crack extension force per unit area of crack from eqn (5). Results are shown in Fig. 6 and it is seen that the K,, value is calculated to decrease as the notch depth increases. Physically, this means that the rate of energy release with crack extension is a function of the initial notch depth. There is support for this in the acoustic emissibn data summarised in Table 1. Noise detected prior to failure has been associated with microcracking and hence energy release in addition to that required to form the main failure crackj IO]. For the 30 unnotched curved beams tested in three-point bending as in Fig. 1, the onset of noise was
A second piece of evidence is available from image analysis studies. Using the technique explained in Section 2, a polished surface parallel to extrusion so containing the most probable fracture initiation sites was examined. All pores having a maximum chord length >0.3 mm were analysed. The largest chord length obtained was 1.6 mm and the average was 0.5 + 0.2 mm. This average cannot be regarded as absolute as it is a function of the lower limit cut off value. In addition, the orientation of large pores is typically perpendicular to the crack growth direction and these pores would thus be relatively ineffectual in acting as cracks. The image analysis of large pores is therefore inconclusive in defining an additional crack depth. Rather it is more likely the case that the additional crack consists of an agglomeration of much smaller pores along the notch root which coalese during the test thus acting as a single crack immediately prior to failure. The polynomial expression of eqn (2) may be further utilised to study the additional crack length. Figure 5 shows how additions to the machined notch depth affect the fracture toughness predicted from this equation. A figure of this type has been deduced by Birch et a/.[91 for a wide range of different graphites. The full line represents the case of an unnotched beam, the K,, value rising rapidly as the assumed additional crack size in-
0.2
0.4
Additional
0.6
0.6
crack
391
1.0
1.2
1.4
1.6
length (mm) -
Fig. 5. K,, values given by the Srawley and Gross polynominal expression as a function of the additional crack length added to the machined defect.
392
A. P. G. ROSE
Symbol Bcam tYP
t.6 t
Notch included K ,c calcutation angle.
x
Curved
60.
SlwleYardoror
l
CUNcd
0.
kawkyandQross
Q
Straight
60.
0
UNed
0.
Srawtqand Bror Critical crack
crtcnsian
force
0.6 P
0
0.1
0.2
Notch
depth
0.4 0.3 I specimen
0.5
0.6
thickness,
x’ -
Fig. 6. The variation of critical stress intensity factors calculated by different methods as a function of notch depth/specimen thickness ratio.
detected at a load of 130 N 2 19 N. This is 15% t 4% of the final failure load and indicates widespread secondary cracking during the test. With a specimen contaming a notch, acoustic emission is detected at a lower load which tends to decrease as the notch deepens, although the results are widely scattered. When the fraction of onset load and failure load is considered, it is evident that cracking starts later in the test as the notch length increases. If this trend is maintained for deeper notches then no cracking will be detected before failure and all the energy release with crack extension during the test will go towards creating the primary crack. To determine the point at which this occurs and hence a lower limit
fracture toughness value requires tests on specimens containing notches of aspect ratio greater than 0.5. Equation (5) predicts that a lower limit G,, is obtained at the minimum value of dp’ldr’. This occurs as x’ + 1 that is as the notch depth approaches the specimen thickness. However, such tests are not easy to interpret due to the proximity of the surface towards which the growing crack propagates. Hence there are considerable experimental difficulties in defining a test to measure a consistent value of critical stress intensity factor using the method of energy release with crack extension. Figure 6 is a plot of critical stress intensity factor K,<, against notch aspect ratio for all small specimen data as
Table 1. Acoustic emission response of notched graphite beams in three-point bend Aspect Ratio of Notch
Load at Onset of Acoustic Emission + 1 Standard Deviation
0
(N)
Fraction of Failure Load at Onset of Acoustic Emission f 1 Standard Deviation
130
f
19
0.15
* 0.04
0.25
80
f
12
0.25
i
0.05
0.36
69
+ 19
0.31
f
0.11
0.48
69
f
0.42
*
0.27
32
393
Calculation of critical stress intensity factors calculated from the two methods described. Following from the discussion above the values obtained from the critical crack extension force are seen to be a decreasing function of notch aspect ratio. Values calculated from the Srawley and Gross polynomial, eqn (2), applied to the 10 mm thick straight beams are also seen to decrease sharply at x’ 2 0.2. In these cases there is 8 mm or less thickness below the notch and the decrease in K,, is associated with a grain size effect[7] as the reduced section is untypical of bulk behaviour. However, all the data from curved beams contains a thickness greater than 8 mm beneath the notch and therefore does not show this effect. The independence to notch depth of K,, values obtained from the polynomial expression is noted, the spread of results over +- one standard deviation being approximately 1.2 +- 0.15 MNm-“I in good agreement with data from whole components[ 11). It appears that the choice of an additional crack length of 0.6 mm below the machined notch compensates well for the decrease in critical crack extension force with increasing notch depth throughout the range examined.
5.
FEASIBILITY
DETERMINATION
OF FRACTURE TOUGHNESS FROM SMALL SPECIMENS OF
IRRADIATED
NUCLEAR GRAPHITE
From the above discussion, the method of fracture toughness determination based on critical crack extension force has been shown to be complex in its interpretation. A further disadvantage is that it requires load and deflection measurements at failure and assumes that deflection at failure is constant independent of notch length. The most promising analysis is clearly the Srawley and Gross approach. This only requires a knowledge of the failure load and additional crack length. For virgin graphite a good agreement between small specimen and bulk K,, values has been obtained by assuming that the material contains an additional crack length of 0.6 mm. However, the question remains of the variation of this additional crack length with irradiation. From the limited amount of image analysis performed it appears that it is not simply controlled by the largest pores present but rather is a more complicated function of the total porosity distribution. Radiolytic oxidation only takes place in the open pores. Hence, image analysis or mercury porosimetry comparing the porosity distributions in virgin and irradiated material are possible techniques to understand any change in the additional crack length with irradiation. Acoustic emission from bend tests will also indicate the onset and extent of microcracking in the fracture of irradiated material which will have a direct effect on the additional crack length. The most immediate method of determining its variation is to perform a series of bend tests using acoustic emission on specimens cut from irradiated components at a range of bum-ups, some samples from each component containing notches of known depth and 0.5 mm root radius. By comparing the fracture loads of notched and unnotched specimens from the same component, a curve to give the additional crack length in the manner of Fig. 5 may be deduced and hence the dependence of the additional crack length on bum-up obtained. In this
manner a consistent method of measuring the variation of K,, with irradiation will be established. Any further work to investigate changes in porosity distribution with bum-up may then be considered. 6. CONCLUSIONS
Measurements of failure loads, deflections and acoustic emission response from notched and unnotched small beams of virgin pitchcoke graphite are reported. Methods of calculating the critical stress intensity factor K,, , are presented and the feasibility of defining a test to obtain K,, values from similar irradiated specimens is discussed. The following conclusions have been drawn. 6.1 A technique based on measurement of the critical crack extension force is complex in its interpretation, requires data on load and deflection at failure and is not considered worthy of further pursuit. 6.2 A consistent method of determining K,, from a polynomial function of failure load and critical crack size has been justified for small beams of a pitchcoke graphite either with or without notches. Values obtained are in agreement with those from tests on whole components. 6.3 In order to determine K,, as a function of irradiation this method requires knowledge of the variation with irradiation of an ‘additional crack length’ below any machined notch. This will be obtained by cutting small specimens from components at a series of known burnups, machining notches of known length into a proportion of the samples from each and testing all of them in threepoint bend. Acknowledgement-This paper is published by permission of the Central Electricity Generating Board. REFERENCES 1.
2. 3. 4. 5. 6. 7.
8. 9. 10. Il.
A. P. G. Rose, L. S. Field and M. 0. Tucker, Exrended Absrracts 16th Biennial Cons. on Carbon, San Diego, 402 (1983). ASTM, E399-78 (1978). British Standards Institute, BS5447 (1977). J. E. Srawley and 9. Gross, ASTM Special Technical Publication 60 I ( 1976). M. 9. Watson, CEGB, Private communication (1983). W. F. Brown and J. E. Srawley, ASTM Special Technical Publication 410,(1966). J. E. Brocklehurst, In Chemistry and Physics of Carbon. (Edited by P. L. Walker, Jr.), 13, 145-283, Marcel Dekker, New York (1974). J. E. Brocklehurst and 9. T. Kelly, Proc. fAEA Conf. Mechanical Behavior of Graphite for HTRs’,Gif-sur-Yvette. IAEA Vienna, 42-50 (1980). M. Birch, R. G. Brown and J. E. Brocklehurst, Exrended Abstracts 16th Biennial Conf. on Carbon, San Diego, 404 (1983). I. M. Pickup, R. G. Cooke and 9. McEnaney, Exrended Abstracts 15th Biennial Conf. on Carbon, Philadelphia, 506 (1981). A. P. G. Rose and M. 0. Tucker, J. of Nuclear Materials, 110, 186-195 (1982).