On the Application of Fracture Mechanics in the ASME
Graphite Core Structural Design Code
Tim Burchell
Oak Ridge National Laboratory
1.
Introduction and Assumptions
The American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel Code
Section III committee established a Project Team on Graphite Core Components (N20070348) to
develop a design methodology for the graphite core components of very high temperature
reactors (VHTR). The incorporation of a fracture mechanics based assessment into the design
methodology is being considered. Here we discuss a possible method of introducing fracture
mechanics into the methodology by linking it to the non-destructive examination (NDE) of
graphite components or billets prior to service. The application of NDE assures the absence of
critical size defects (surface or bulk depending upon the NDE technique applied) in components
where the critical defect size has been established using fracture mechanics principals and the
calculated peak stress. The proposed approach is essentially a proof or screening test (NDE)
designed to exclude components containing “disparate” flaws that are responsible for the low
strength tail of the (Weibull) distribution. The application of NDE as an in-service inspection
tool is not discussed here.
To perform fracture mechanics assessments of nuclear graphite the following assumptions are
made:
1. Graphite behaves in a linear elastic manner, this is reasonable true for fine grained
graphites, but is not the case for coarse grained graphites, where significant process zone
activity occurs (non-linear stress-strain curve).
2. A standard method for determining the Critical Stress Intensity Factor, KIc, exists (ASTM
is currently developing such a test method).
3. The designer can predict the peak stress within a component during its lifetime.
4. The strength of the graphite is known during its lifetime (environmental effects must be
considered).
5. NDE methods exist to screen components or graphite billets to assure the absence of
defects greater than a specified size. Ultrasonic methods are discussed here, but other
techniques may be applicable, e.g., eddy-current testing.
6. Fracture mechanics assessment cannot be applied to very small defects which are similar
in size to the grain size since it has been shown to be insufficiently conservative at small
crack lengths [1].
1
2.
Background
The critical stress intensity factor for mode I crack opening is given by
KIc = σ√πc (MPa.√m)
(1)
where c = crack half length, m
σ = applied tensile stress, MPa
From equation (1), for a given value of KIc and a known stress level, the size of the critical defect
may be estimated. Alternately, if a component can be screened non-destructively to assure that
defects above a certain size do not exist, and knowing KIc for the material, the maximum stress
that can be tolerated may be estimated. These arguments are further elucidated below.
The minimum theoretical resolution limit for graphite via an ultrasonic method can be calculated
from
λ = c/ (m)
(2)
where λ = wavelength of sound in the graphite, m
 = the sound wave frequency, Hz
c = velocity of sound in graphite (~2600 m/s)
For graphite the velocity of sound is 2000-2800 m/s, thus for a 0.5 MHz frequency the
wavelength is ~5.2 mm. Smaller flaws may be resolved only by using higher frequencies
(resolution is ~2.6 mm at 1.0 MHz) but the attenuation increases as the frequency increases, and
therefore the greater resolution obtained at higher frequencies may be negated by the limited
penetration depth (range). Experiments on graphite have shown that a frequency of 0.5 to 1
MHz represents an acceptable compromise.
3.
Discussion
3.1
Determining Critical Stress Intensity Factor, KIc
Several experimental techniques have been developed for measuring the critical stress intensity
factor, KIc, of graphite. However, the measured value of KIc has been determined to be very
sensitive to the experimental method. In a comparative study of a fine-grained, isotropic
graphite, Burchell, Oku, and Eto [2] determined KIc using five different techniques. The mean
value of KIc varied from 0.83 MPa√m for the centrally slotted disc specimen geometry to 1.16
MPa√m for the chevron notched short rod specimen geometry (Table 1). Specimen geometries
which allow slow crack growth were observed to yield higher values of KIc.
2
KIc, MPa.m^(1/2)
Table 1. A comparison of KIc data obtained from different test geometries in IG-110 graphite [2]
Specimen geometry
KIc (MPa√m)
Laboratory were
tests performed
Mean
St. deviation
Chevron notched short rod
1.16
0.05
ORNL
Disc compact tension
1.12
JAERI
Cylindrical bend specimen
1.07
JAERI
Short rod
0.94
JAERI
Compact tension
0.92
0.04
JAERI
Centrally slotted disc
0.83
0.03
ORNL
Romanoski and Burchell [3] applied the chevron notched short rod geometry in a study of the
effect of specimen volume on the KIc of three graphites. They observed that the measured value
of KIc increased with specimen volume for all the graphites examined.
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
H-451
S-2020
IG-110
10
15
20
25
30
CNSR Specimen Diameter, mm
Fig. 1. The variation of KIc with chevron notched short rod volume for several graphites
These data highlight the need for a standard method for the measurement of KIc in graphites.
Moreover, if toughness is to be a factor in the selection or design methodology it is vital that self
consistent data is used for the graphites being considered, i.e., that the KIc is determined in
accordance with a reproducible standard method and from specimen geometries and volumes
that are appropriate for the graphite being tested. Because of these requirements an ASTM
standard test method for the determination of graphite KIc is currently in preparation.
3.2 Small Flaw Considerations
As discussed by Romanoski and Burchell [1] the application of fracture mechanics to small flaws
in graphite can lead to non-conservative estimated of the failure stress. The two plots in Fig 2
are for grade H-451 graphite and show (i) the dependency of the fracture stress on the defect half
length (calculated from eq. 1), and (ii) the mean strength of H-451 graphite from tensile tests. In
Fig. 2, the area above both lines represents the failure region, where both the fracture mechanics
3
criterion and maximum principal stress failure criterion (MPSFC), based on tensile strength,
predict failure. The area below both plots represents the non-failure region, where neither
criterion predicts failure. The two plots intersect at a defect half length of 2.4 mm. The region at
small flaw lengths, bounded by the two plots in Fig 2 is labeled non-conservative, and represents
the small flaw region (at defect half lengths < 2.4 mm) where fracture mechanics should not be
applied because it overestimates the fracture stress. Essentially, in this region, fracture is
controlled by intrinsic defects. The region between the plots at flaw lengths > 2.4 mm represents
the region where the maximum principal stress failure criterion (based on tensile strength)
overestimates the fracture stress and where a fracture mechanics assessment should be
considered as applicable. The cross over point at 2.4 mm is very dependant upon the value of the
tensile strength (or minimum ultimate strength), and thus it is particularly important that the
selected value be statistically relevant.
100
Kic = 1.3 MPa.m^(1/2)
Tensile Strength = 15 MPa
Critical Defect Half Length
Mean Tensile Strength
MPSFC
overestimates
failure stress
Half length = 2.4 mm
Stress, MPa
Non conservative
10
KIc = σ.√πc
1
0.1
1
10
100
Defect half length, mm
Fig. 2. Fracture strength versus crack length diagram for H-451 graphite illustrating the
failure/non-failure boundary for fracture, and the small flaw limit for application of a fracture
mechanics failure criterion.
4
3.3
Ultrasonic NDE Experience
Prior work on the application of ultrasonic NDE to graphites is reviewed. Ultrasonic inspection
of billets is routine during manufacture and has in the past been successfully applied to finished
components.
3.3.1 Toyo Tanso/HTTR Project
Ultrasonic NDE methods were developed to inspect core components for the High Temperature
Test Reactor (HTTR). A 5 mm diameter artificial flaw was introduced to a series of test blocks
of grade IG-110 graphite. The artificial defect was detected to a depth (range) of 550 mm with a
signal-to-noise (S/Nmax) ratio of >4 using a pulse-echo technique (Fig. 3) [4]. Fig. 4 shows
attenuation as a function of range (distance to the defect) [4]. The attenuation coefficient can be
estimated for IG-110 graphite to be ~0.06 dB/mm. Sumita et al established an acceptance
criterion whereby the signal returned from a defect, or defective area (echo) should be < 4Nmax,
where Nmax is given by:
Nmax = C – (Dx)
where x = range (distance from surface to defect), mm
d = attenuation coefficient, dB/mm
C = Noise-echo height at the block surface.
Fig. 3. Ultrasonic flaw detectability of grade IG-110 graphite
5
(3)
Fig. 4. Noise echo level as a function of range for grade IG-110 graphite
3.3.2 GrafTech (UCAR) experience
GrafTech International Ltd has developed an automated ultrasonic NDE system for inspecting
graphite blocks. Shives [5] reports the ultrasonic velocity in graphite ranged from 1800 m/s to
2800 m/s and the minimum detectable flaw varied from 2 to 20 mm depending on probe
frequency for pulse-echo testing. The GrafTech automated system is based on through
transmission rather than a single probe (pulse-echo) system. GrafTech successfully correlated
sonic velocity with modulus of rupture [5].
3.3.3 ORNL experience
Ultrasonic inspection has been widely used at ORNL for the NDE of graphite slabs. Kennedy
reported the attenuation coefficient of H-451 was 0.145 dB/mm (axial) at a frequency of 0.5
MHz (determined through a 50 mm thick section of graphite) [6]. Data for several isostatically
molded, fine-grained graphites measured at an ultrasonic frequency of 0.5 MHz are shown in
Fig. 5 [7]. The attenuation coefficient for IG-110 is in agreement with at reported by Sumita et
al [4]. The attenuation coefficient of grade 2020 is similar to IG-110, whereas the coefficient for
grade TS-1792 was slightly greater that that of grades IG-110 and 2020. Penetration depths in
excess 1.2m were achieved with attenuations of ~60 dB for the isomolded grades tested.
6
Fig. 5. Average 0.5 MHz frequency ultrasonic attenuation in three graphites [7]
In a study of anisotropic, coarse grained graphite (grade AGOT), Kennedy and Kehne [8]
showed the attenuation coefficient varied with orientation with respect to the extrusion axis from
~ 0.5 dB/mm parallel to the extrusion axis to ~1.3 dB/mm perpendicular to the extrusion
direction.
3.3.4 UK/CEGB Experience
Ultrasonic NDE was satisfactorily applied to grade IM1-24 graphite. Holes of 3mm diameter
were detected to a depth (range) of ~70 mm with a signal-to-noise ratio of greater than two using
a 1MHz frequency probe. The signal attenuation was measured at ~60 dB giving an attenuation
coefficient of ~ 0.17 dB/mm at 1 MHz [9].
Bragoli et al [10] reported the attenuation coefficient of several graphites as a function of
ultrasound frequency. The attenuation of grade IM1-24 increased approximately as the first
power of frequency in the range 1 to 5 MHz. The rate of increase of attenuation coefficient with
frequency was less significant for micro fine grain ZXF-5Q, but was very significant for a coarse
grain electrode graphite, where frequencies > 2 MHz could not be used.
7
3.3.5 Ultrasonic NDE Summary
Based on the forgoing discussion, ultrasonic inspections can be successfully applied to graphites
to detect flaw and regions of high attenuation. The size of the flaw that can be detected is a
function of the ultrasound frequency, with smaller flaws detected at higher frequencies. The
range at which flaws may be detected depends upon the graphites attenuation coefficient, which
is dependant upon the graphite texture (increases as the texture coarsens), and the frequency
(increases with frequency). For fine and medium grain graphites detection of 4 to 10 mm sized
flaws at reasonable range (90-200 mm) with signal-to-noise ratios >2 would appear to be
feasible. Several researchers have correlated the signal attenuation directly with strength [4] or
have derived a strength parameter (attenuation/modulus) which was reported to correlate with
strength [7, 11]. When applying NDE methods consideration must be given to the orientation of
potential defects, i.e., to the direction of the principal stress, and to the NDE probe being used.
Defects that are oriented perpendicular to the principal stress are significantly more damaging
than those aligned parallel to the stress. Moreover, a defect aligned perpendicular to the
ultrasound propagation direction is considerably easier to detect than one with off axis
alignment.
3.4
Combining Fracture Mechanics and Ultrasonic NDE
Fracture mechanics data calculated from eq. 1 for several hypothetical graphites are shown in
Figs. 6 and 7. In Fig. 6 the flaw tolerance of three graphites whose KIc values are 0.8, 1.0, and
1.2 MPa√m, respectively, is reported. Fig. 7 shows the critical defect half size as a function of
KIc for three tensile stress level, 4, 8, and 12 MPa. In both Figs the defect detection limit via
ultrasonic inspection is shown for frequencies of 1 and 0.5 MHz (calculated from eq. 2). Note
the indicated defect size is the critical defect half length, so the defect detection limit is also
plotted as the defect half length in Figs. 6 and 7.
8
Critical Defect Half Size, mm
Flaw Tolerance of Graphite as a Function of KIc
100.00
KIc = 0.8 MPa√m
KIc = 1.0 MPa√m
10.00
KIc = 1.2 MPa√m
1 MHz Detection limit/2
1.00
0.5 MHz Detection
limit/2
0.10
4
6
8
10 12 14 16 18 20 22
Applied Tensile Stress (MPa)
Fig. 6. The variation of critical defect half size with applied stress as a function of critical stress
intensity factor, KIc
From Fig. 6, if graphite can be ultrasonically inspected at a frequency of 1 MHz to assure that no
defect above a size of 2.6 mm (1.3 mm half length) are present, then the advantage of a tougher
graphite may be quantified for that flaw size as shown in Table 2.
Table 2 Predicted failure loads for three hypothetical graphites at two critical defect sizes
Predicted failure load (MPa)
KIc (MPa√m)
1.0 MHz (2.6 mm defect)
0.5 MHz (5.2 mm defect)
0.8
12
9
1.0
15
11
1.2
18
13
The data in Table 2 demonstrates the potential benefit of using a graphite grade with a higher KIc
value. As KIc increases the predicted failure stress for a 2.6 mm defect increases. However,
graphites with higher KIc typically exhibit lower tensile strengths than graphites with lower KIc
values. Additionally, ultrasonic attenuation may be greater in the tougher graphites - making
flaw detection more challenging. Where greater penetration depths are required a lower
frequency may be used. For a 0.5 MHz probe (defect detection limit is ~ 5.2 mm, corresponding
to a crack half length of 2.6 mm) the predicted failure stresses are 9, 11, and 13 MPa for KIc
values of 0.8, 1.0, and 1.2 MPa√m, respectively (Table 2). Again the benefit of tougher graphite
is apparent.
This is quantified slightly differently in Fig. 7, where the defect size for tensile stresses of 4, 8,
and 12 MPa are shown as a function of KIc. At an applied stress of 4 MPa, ultrasonic NDE at 0.5
MHz assures that no critical size defect exist and failure is not predicted for graphite with KIc
9
values between 0.8 and 1.2 MPa√m. The 12 MPa applied stress line falls partly below the 0.5
MHz detection limit for lower KIc values. If NDE can assure no defects >2.6 mm (1 MHz) are
present then 8 and12 MPa peak stress can be tolerated for all KIc values (although the 0.8
MPa√m case at 12 MPa is marginal). However, if the graphite is subjected to NDE at 0.5 MHz
and the absence of flaws > 5.2 mm can only be assured, a stress of 12 MPa can only be tolerated
if KIc > 1.1 MPa√m and a stress of 8 MPa can be tolerated for all KIc values considered. Again
the advantage of a tougher graphite grade is apparent since it can tolerate a larger flaw at a given
stress.
Thus for a specific example of a graphite with a KIc = 1, and an end of life stress of 12 MPa, the
component must be inspected to assure that it does not contain a defect with half lengths greater
than ~ 2.2 mm (i.e., defect lengths < 4.4 mm) failure is probable at that peak stress. Similarly, if
the graphite has KIc of 1 and an end of life stress of 8 MPa the component must be inspected to
assure it does not contain a defect greater than ~10 mm.
Critical Defect Half Size as a Function of KIc for
Graphite
Stress = 4 MPa
Critical Defect Half Size, mm
30.00
Stress = 8 MPa
25.00
Stress = 12 MPa
20.00
1 MHz Detection
Limit/2
0.5 MHz Detection
Limit/2
15.00
10.00
5.00
0.00
0.7
0.8
0.9
1
1.1
1.2
1.3
Critical Stress Intensity Factor, KIc (MPa.m^1/2)
Fig. 7. The dependence of critical defect size on KIc for three stress levels
10
4.
Examples
Table 3 reports the certain physical properties of two graphites. The fracture toughness was
measured using the chevron notched short rod technique on 12.5 mm diameter specimens [3].
The mean tensile strength is taken from the literature. For both graphites two peak stresses are
considered, 8 MPa and 12 MPa. The design margin (mean tensile strength minus the peak stress)
is reported towards the bottom of Table 3. The critical defect size is calculated from equation 1
using the tabulated KIc and peak stress. Moreover, for each grade, the minimum detectable flaw
size, attenuation, signal attenuation limit, and predicted range (maximum distance from the
surface that a defect may be detected) are reported at two probe frequencies (0.5 MHz and 1
MHz).
Table 3. Comparison of properties and key ultrasonic NDE parameters for two graphites
GRAPHITE A: Fine
GRAPHITE B: Medium
Property or Parameter
grain, iso-molded
grain, extruded
KIc, MPa√m
0.98
1.35
Tensile Strength*, MPa
24.5
15
Peak Stress, MPa
8
12
8
12
Defect half size, c, mm
4.78
2.12
9.06
4.03
Critical defect size, mm
9.55
4.25
18.13
8.06
Probe Frequency, Hz
0.5
1.0
0.5
1.0
Attenuation, dB/mm
0.06
0.12
0.145
0.29
Signal Attenuation Limit, dB
50
50
50
50
Pulse-echo range, mm
417
208
172
86
Probe resolution, mm
5
2.5
5
2.5
Margin (TS-PS)
16.5
12.5
7
3
Critical defect size/resolution
1.9
1.7
3.6
3.2
* Strength changes due to irradiation damage are not considered in this example
The attenuation is greater for the medium-grain, extruded graphite, thus limiting the range over
which defects may be detected. However, the critical defect size is significantly larger in the
medium grained, extruded graphite by virtue of its greater flaw tolerance (larger KIc). A useful
parameter is the ratio of the critical defect size (2c) to the resolution (detection limit) for the
particular ultrasound frequency being used. Ideally, this ratio should be as large as possible
consistent with achieving the desired range (which decreases as frequency increases).
Careful inspection of Table 2 indicates that both graphites offer certain advantages, but also have
certain disadvantages. While graphite A offers a larger margin compared to graphite B, the size
of the critical defect is almost twice as large in graphite B as in graphite A, perhaps offering
greater ease of detection, as indicated by the larger values of the ratio (2c/resolution). However,
graphite B has greater attenuation than graphite A, limiting the range at which defects may be
detected. These differences are summarized in Table 4.
11
Table 4 Graphite comparison summary
Graphite
Critical defect
Margin (mean
size, 2c,
strength – peak
stress)
A
Smaller
Greater
B
Greater
Smaller
Detection Range
Greater
Smaller
Detectability
Ratio
(2c/resolution)
Smaller
Greater
The advantage of a greater detectability ratio is offset to some extent by the smaller range of the
medium-grain, extruded, graphite B. Ultimately, graphite selection becomes a trade-off between
the increased margin offered by the higher strength graphite and the increased flaw tolerance of
the tougher (greater KIc) graphite. A further complicating factor is the degree of confidence to
which the strength of the selected graphite is known. It may be unsatisfactory to use the mean
strength if the strength distribution is very wide, in which case the design strength may be
significantly less than the mean strength.
5.
Methodology
The following is a suggested design assessment methodology:
1. Determine the peak stress in a component at end of life.
2. Determine the specified strength of the graphite, i.e., mean tensile strength or specified
minimum strength (environmental effects may have to be accounted for).
3. Determine adequacy of margin for safety class, if sufficient, then
4. Determine the KIc value for the candidate graphite in accordance with the recommended
ASTM standard.
5. Calculate the critical defect size (c) for the graphite at the stress level determined in step
1 and ensure this is > small flaw limit (SFL).
6. Establish the minimum detectable flaw size for the NDE method being employed and the
detection range for the selected graphite
7. Assure the ratio (2c/resolution) < 2 and the detection range is adequate for the
components design.
8. Complete NDE of component/billet and assure defects are smaller than the critical size
calculated in step 5 (2c).
A flow chart illustrating this method is shown in Fig. 8.
6.
Conclusions
The combined application of Ultrasonic NDE and LEFM to graphite has been discussed. The
small flaw limit above which fracture mechanic can be applied was reported and the benefits and
limitations of a tougher (more flaw tolerant) graphite was demonstrated. The relationship
between the flaw detection limit and range, and their dependency on frequency was discussed.
12
These interdependencies were illustrated by comparing the behavior of two graphites. Finally, a
recommended methodology for performing a combined fracture mechanics and NDE assessment
was outlined. The application of this methodology to graphite core components would reduce
the likelihood of components contain “critical” defects entering service.
7.
References
1. G. R. Romanoski and T. D. Burchell, Fracture in Graphite. In Carbon Materials for
Advanced Technologies, ed. T. D. Burchell, Elsevier Science Ltd, Oxford, 1999, pp.485534.
2. T. D. Burchell, T. Oku, and M. Eto, A comparison of fracture toughness measurement
techniques as applied to nuclear graphite. In Proc. CARBON’ 90, pp. 278-279, 1990.
3. G. R. Romanoski and T. D. Burchell, Specimen size effect on fracture toughness of
nuclear graphite. In Proc. 20th Biennial Conference on Carbon, pp.584-585, 1991.
4. J. Sumita, T. Iyoku, M. Ishihara, and T. Konishi. Acceptance test and ISI program for the
HTTR core components. In. Proc INGSM-4, Marugame, Japan, 2003.
5. G. Shives. Non-destructive testing of graphite materials, In. Proc. INGSM-3, Parma,
USA, 2002.
6. C. R. Kennedy. Tensile testing grade H-451 graphite billets R58-854 and R58-394,
ORNL/TM-11045, August 1990.
7. C. R. Kennedy. Statistical characterization of three grades of large billet-graphites:
Stackpole 2020, Union Carbide TS1792, and Toyo Tanso IG11, ORNL/TM-10457,
September 1987.
8. C. R. Kennedy and M. T. Kehne. Fracture toughness of anisotropic graphite, In Proc.
17th Biennial Conf. on Carbon, pp. 289-290, 1985.
9. T. D. Burchell. Ultrasonic studies of a nuclear graphite, In Proc. CARBON ’88, pp. 552554, 1988.
10. P. B. J. Bragoli, D. P. Almond, and B. McEnaney. Ultrasonic attenuation in graphite,
CARBON 26(1), pp. 108-110, 1988.
11. C. R. Kennedy. Evaluation of fracture strength by sonic testing, In. Proc. 15th Biennial
Conf on Carbon, pp. 536-537, 1981.
13
Determine
peak stress
Determin
Determine
e
specified
specified
minimum
minimum
strength
strength
Sufficient
margin or
sufficiently
low failure
probability?
Increase margin or
reduce failure
probability
Determine KIC and
calculate “c” at
peak stress, >SFL?
Determine detection
limit and range
(functions of
ultrasonic frequency
selected)
Select tougher
graphite or reduce
frequency
(2c/dl) < 2
Reduce
dimensions or
reduce
frequency
Sufficient
range?
Complete NDE & assure defects size < 2c
END
Fig. 8. Methodology flow chart for graphite component integrity assessment
14