Calibration for Machine Deformation for High Temperature Fracture Testing
S.K. Ray and G. Sasikala
Materials Technology Division, Indira Gandhi Centre for Atomic Research,
Kalpakkam 603 102, India
[email protected]
ABSTRACT
Quasi-static fracture toughness testing at elevated temperatures can be simplified to a great extent if direct measurement of
load line displacement with COD gages can be avoided. Towards this, a procedure has been developed for calibrating the
non-linear elastic machine deformation as a function of load, using the data for a blank specimen deforming in the elastic
regime. The calibration procedure is demonstrated for two temperatures, viz., 653 K and 803 K. The viability of using the
calibration method for evaluating the elastic plastic fracture toughness parameter J
for a modified 9Cr-1Mo steel is
established for the two elevated temperatures.
Introduction
Structural integrity assessment of components requires the elastic plastic fracture toughness property of the material of the
component at the service temperature. For example, for “leak-before-break” analysis, the J-resistance curve for the material
needs to be generated to different extents of crack growth depending on the component. The value of elastic plastic fracture
toughness J for 0.2 mm crack extension, J0.2 , for the modified 9Cr-1Mo steel is required at 653 and 803 K. Experimental
evaluation of J -R curves requires simultaneous load (P), load line displacement (v) and crack extension (∆a) data, and J
testing in servohydraulic/electromechanical machines can be cumbersome when high temperature extensometers are used for
measuring load line displacement (LLD), particularly if potential drop method is to be used for online crack length
measurement. ASTM E1820-01 standard [1] prescribes for fracture tests with bend specimens, that (i) when LLD is measured
at a remote point, corrections must be made for excluding the elastic deformation in the loading train and the plastic
deformation (brinneling) in the specimen at the load points (section 8.3.1.1), and (ii) when LLD is obtained from measurements
on a plane not containing the load line, ability to infer LLD to within 1% accuracy must be demonstrated (section 8.7.2). For
compact tension (CT) specimens, in absence of specific recommendations, the same criteria can be used as guidelines. This
paper reports the implementation of a calibration method for inferring LLD from actuator displacement for CT specimens tested
in monotonic ramp loading under actuator stroke control.
Calibration Scheme
For a fracture test with a pre-cracked specimen, the actuator displacement (stroke) is given by
xt = xm + xsp
where
(1)
x t = total actuator stroke
xsp = (elastic + plastic) displacement of the specimen
x m = the machine deformation
Here,
xsp is required for computation of J for the specimen. The “machine” is defined as the entire load train assembly
between the fixed cross head and the moving actuator, excluding the specimen. The brinnelling displacement in the specimen
adjacent to the loading pins is considerably smaller than x m for typical testing configurations and can be ignored. xm can be
considered to be elastic and is a strongly non-linear function of P .
The calibration procedure proposed here involves determining the x m − P relation, from the x t − P data for a specimen with
known elastic compliance C (as measured at the loading pins) deforming in the elastic regime, using the relation
x m (P ) = x t − C ⋅ P
(2)
Similar calibration procedure has been used [2] for inferring gauge extension during tensile testing from the stroke data in
screw-driven machines. Since x m would depend on the temperature profile on the elements of the loading train, it is
necessary to carry out this calibration for each temperature level of fracture testing, using the same testing configuration as for
the fracture tests, and ensuring that the calibration covers the maximum load levels for the fracture test data. Calibration
testing proposed here consists of three steps. First the blank specimen is ramped at the chosen constant actuator stroke rate
to the desired load level. The contact with loading pin causes brinneling in the specimen in the adjacent regions. Therefore the
second step consists in arresting the actuator motion for a period sufficient to complete any load relaxations in the brinnelled
regions. It is necessary to ensure that the maximum load at the end of the relaxation step covers the maximum loads for the
fracture tests. In the third step, the specimen is ramped down to zero load at a suitable actuator speed. The second step
ensures that deformation in the third step (unloading) is fully elastic.
As a part of a campaign for determining the fracture toughness of a modified 9Cr-1Mo steel (P91 grade) at 653 and 803 K, the
calibration was carried out at these two temperatures, using blank specimens of the same material. The external dimensions of
the blank calibration specimen, Fig. 1(a), and that for the elevated temperature fracture tests, Fig. 1(b), were kept identical to
minimize differences in temperature profiles on the load train for the calibration and the fracture tests. Still, typical x t value at
the end of a fracture test is of the order of a few mm, compared to ~ 0.5 mm or less for a calibration test. Consequently, the
temperature profiles of the loading trains for these two cases can not be strictly identical. The resultant error in the calibration
however is not expected to be significant.
a
a
φ12.7±0.01
φ12.7±0.01
b
R6
17.6
±0.01
Wire cut
2
6
17.6
60
60
17.6
6
3±0.01
17.6
50
62.5
50
62.5
Figure 1. The specimen design used for (a) machine calibration and (b) fracture tests.
The dimensionless compliance E′BC for the specimen was determined as 5.55, from 2-D elastic boundary element method
(BEM) computations carried out for one symmetric half of the specimen using the BEM code of Portela and Aliabadi [3] with
marginal modifications. Details of the computations are not presented here and may be found in another paper [4]. This value
of E′BC computed here for a/W = 0 is consistent with the results reported by Saxena and Hudak [5].
Experimental
The calibration and fracture tests were carried out in a servohydraulic universal testing machine using the same testing
assembly. The specimen and the high temperature pull rods were enclosed in a 2-zone resistance heating furnace, and the
specimen temperature was held within ±2 K of the desired value. All loading/unloading for the calibration tests were carried out
at a constant actuator speed of 0.1 mm/min. For the fracture tests with pre-cracked specimens, the LLD gage used was double
cantilever type with quartz arms transferring displacement (through a port in the furnace) to a metallic foil strain gauge
maintained at ambient temperature outside the furnace. To avoid any torque on these quartz rods as the specimen deformed,
only one of the quartz arms had a V-groove that sat snug on the integrally machined knife-edge on the specimen; the other
edge pressed lightly against the flat surface of the specimen opposite to the knife-edge. The P, x t , and also the LLD data for
the fracture tests, were captured with 6½ digit digital voltmeters using appropriate hardware and software. The loading for the
fracture tests were the single ramp type. The calibration specimens were made of 9 Cr-1Mo steel in normalized and tempered
(NT) condition, whereas the fracture tests were conducted on modified 9Cr-1Mo steel in NT without/with subsequent thermal
ageing (2900 h/ 923 K) meant to simulate long term service exposure.
Calibration Results and Validation
The P − x t data for any unload, after correcting for the permanent offset due to brinneling, can be used to determine the
machine calibration equation using Eq. (3). Provided the tabulated x m − P data have been collected at sufficiently close
intervals and cover the maximum loads for the fracture tests, these could be directly used in analyzing the fracture test results,
using linear interpolation when needed. It is practically more convenient to use an empirical functional form, and also use least
square fitting procedure for simultaneously correlating data from several unloads for improved confidence. Because P − xt
data showed strong curvilinear variation at low P, only data above a small cut off load (determined by trial and error) were
used for the least square fit; this considerably simplified the functional form. For the present calibration campaign, for a given
calibration temperature, data for all unloads were least-square fitted to the following empirically chosen functional form:
xt = δ i + C ⋅ P + xm (P ) = δ i + C ⋅ P +
∑q
j
⋅P
rj
(3)
j
where
and the set
δ i = the zero-offset for i-th unload,
(q j , r j ) = the fitting constants
Note that x m = 0 for P = 0 . The values for fitting constants were determined using least squares optimization, and the
number of (q j , r j ) pairs was chosen so as to restrict maximum errors of fit to acceptable values. It is quite possible that
functional forms alternative to Eq. (3) may prove better for other testing systems. Once the x m (P ) equation for a specific test
temperature is obtained, it can be used for inferring x sp from the x t data for a fracture test at the same temperature. Since
the computed x m (P ) equation is valid only above the lower cut off load, x sp thus inferred for fracture test has an offset at
P = 0 . Since for fracture tests, the early P - inferred x sp data are linear well beyond the cut off load, these can be easily backextrapolated to P = 0 to determine this offset value, which can then be used to correct the inferred data to ensure that x sp
= 0 for P = 0 .
Figure 2. The P - xt data for the calibration test on a 10 mm thick blank specimen at 653 K. (a) Data for three load-relaxunload cycles (b) Low load data for the loading and unloading segments of the first cycle.
The fracture test data, where measured xsp data were also available, enabled direct evaluation of the viability of this
calibration procedure. A typical plot of the P − x t data is shown in Figure 2(a) for the calibration test on a 10 mm thick
calibration specimen for three load-relax-unload cycles at 653 K. At the resolution of this figure, the permanent offsets due to
brinneling, or the hysteresis between loading and unloading, are not discernible. Figure 2 (b) shows the low-load segments for
the loading and unloading of the first cycle. From this plot, a permanent offset due to brinneling is estimated as ~ 25 µm. The
best fit was obtained (with x m in mm, P in N), as
xm (P ) =
∑ q ⋅ (P 10 )
j
4 rj
(4)
2
using a cut off load of 80 N for 653 K and 803 K. The scaling of P was necessary for better numerical conditioning. Figures
3(a-b) show the errors of fit to eq. (3) for these test temperatures; the values for the optimized constants are not reported here.
The error of fit beyond the cut-off load was within ± 2 µm for 653 K, and for 803 K within ± 4 µm . These errors are quite
comparable to typical resolutions for high temperature LLD extensometers.
Figure 3: The errors of fit of the P − x t data to eq. (3) for (a) 653 K and (b) 803 K. Different symbols indicate data for different
unloadings.
Several fracture tests both at 653 K and 803 K have been carried out [5] on modified 9Cr-1Mo steel (P91) in NT and aged
conditions. Here only limited results are presented to illustrate the points involved. Typical plots of P versus x sp , both
measured and inferred, are shown in Fig. 4(a-b) for NT and aged materials respectively at 653 K. The tiny load jumps in Fig.
4(a) might be because of minor pop-in crack extensions leading to tiny slippages of the flat-ended arm of the LLD gauge. The
agreement between the measured and inferred x sp was better than the 1%.
Figure 4. Typical plots of P versus x sp , both measured and inferred, for tests at 653 K, for (a) NT and (b) aged materials.
A typical P versus x sp plot for the NT material tested at 803 K is shown in Fig. 5(a). The encircled region in this figure, shown
in expanded scales in Fig. 5(b), corresponds to a slippage of the LLD gage (not associated with any pop-in) resulting in a
backward displacement. It may be noted that, beyond this point, the measured x sp continuously increased and eventually
exceeded the inferred x sp . This suggests that beyond some critical rotation in the specimen, the free arm of the gage started
sliding down the flat surface. This actually shows that inferred x sp , rather than x sp measured by the specific specimenextensometer configuration used here, should be more reliable. Also, even for large displacements, unlike measured x sp ,
inferred x sp does not require to be corrected for the LLD measurement line progressively shifting away from loading line (cf.
the rotation correction for crack length calculation from compliance data, Section A2.4.4 in ASTM E 1820).
11400
(b)
11300
11200
P, N
11100
11000
P91, N&T, 803 K
measured
LLD:
inferred
10900
10800
1.40
1.45
1.50
1.55
1.60
1.65
1.70
xsp, mm
Figure 5: (a) A typical plot of P versus x sp , both measured and inferred, for the NT material tested at 803 K. (b) magnified
view of the encircled region in (a), corresponding to a slippage of the LLD gage resulting in a backward displacement.
The nominal J (without crack growth correction) resistance (Jnom-R) curves for 653 K computed using inferred and measured
x sp are presented in Figs. 6 (a-b) respectively. Data for both NT and aged conditions are included in these plots and are not
distinguished. It is clear that use of inferred x sp results in considerably less scatter compared to measured x sp reflected in
900
900
n
800
700
n
Fit: J = A(∆a)
A = 462 (± 20)
n = 0.57 (± 0.07)
800
700
600
500
500
400
400
nom
Jnom, kJ.m
-2
600
Fit: J = A(∆a)
A = 488 (± 33)
n = 0.53 (± 0.13)
300
200
100
(a)
P91 653 K (inferred lld)
ASTM Multi-specimen method
300
200
100
(b)
P91 653 K (measured lld)
ASTM Multi-specimen method
0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
∆a, mm
∆a, mm
Figure 6. Results of multiple-specimen analysis of fracture test data for 653 K following ASTM E1820-01 prescriptions (a)
inferred LLD and (b) measured LLD.
the fact that the constants of fit of the J − ∆a relation, J = A ⋅ ∆an , A and n are better determined (indicated in these figures).
2
Also, estimate of J 0.2 (~290 kJ.m− ) is obtained using inferred x sp is conservative compared to using measured x sp (~330
2
kJ. m− ). Identical conclusion was reached by comparing the Jnom-R curves obtained using inferred and measured x sp at 803
K. These results however, are not presented here. As can be readily anticipated from these results, similar conclusions could
be drawn by comparing inferred and measured x sp when using single specimen normalization method for J estimation (cf.
A15, ASTM E1820); these results are described in Ref [6] and are not presented here.
Conclusions
The calibration procedure described in this paper is simple, and proved to be adequate for determining J resistance (Jnom)
curves by the multiple specimen method that uses monotonic ramp loading tests up to 803 K in the case of 9Cr-1Mo steel.
Indeed, x sp inferred from x t using this calibration procedure is to be preferred to that measured by high temperature LLD
gages when there is a possibility of gage slippage, e.g., due to pop-in crack extensions. This success can be attributed to the
fact that the error due to ignoring brinnelling in the specimen close to the loading pins, which is not accounted for by the
calibration method, was small enough to be ignored for the high toughness material. The calibration procedure, however, does
allow an approximate estimate of the extent of brinnelling and the resultant maximum error in J estimation, which should be
useful in planning the tests.
References
1.
Standard Test Method for Measurement of Fracture Toughness, ASTM E 1820-01, American Society for Testing of
Materials (ASTM), Philadelphia (2001).
2.
Ray, S.K., Bhaduri, A.K., and Rodriguez, P., “Calibration of Machine Deformation for Screw Driven Machines without
Gauge Length Extensometry,” Trans. Ind. Inst. Metals, vol 46, 71-75 (1993).
3.
Portela, A., and Aliabadi, M.H., “Crack Growth Analysis using Boundary Elements”, Computational Mechanics
Publications, Southampton(1993).
4.
Ray, S.K., and Sasikala, G., “Calibration of Machine Deformation for High Temperature Fracture Testing under
Quasistatic Loading Conditions,” communicated to Int J Fracture (2006).
5.
Saxena, A.K., and Hudak, S.J., “Review and Extension of Compliance Information for Common Crack Growth
Specimens”, International Journal of Fracture, vol. 14 453-468 (1978).
6.
Sasikala, G., and Ray, S.K., “Effect of Ageing on Fracture Toughness of a Modified 9Cr-1Mo Steel Plate at Service
Temperatures”, in Proc. National Conference on Ageing Management of Structures, Systems and Components,
NCAM 2004, Nuclear Power Corporation of India Ltd, Mumbai 15-17 December, 2004, Paper A-15 (2004).