AN IMPROVED STRAIN GAUGE
METHOD FOR MEASURING K I D FOR
A PROPAGATING CRACK
R. J. SANFORD Department ofMechanica1 Engineering, University of Maryland, USA.
J. w.DALLY
J. R. BERGER
Department of Mechanical Engineering, Unioersity of Maryland, U S A .
National Institute of Standards and Technology, USA.
An improved strain gauge method for measuring the dynamic stress intensity factor, K,,, of a running crack is
described. By orienting the gauge relative to the crack propagation path the gauge response is optimized for the
analysis. Higher order terms in the dynamic strain field representation are demonstrated to be important in the
analysis. An application of the method is illustratedin the measurement of K,, for a very hard alloy steel, 4340, with a
crack propagatingat 656 m/s.
where
1 INTRODUCTION
The dynamic fracture analysis of engineering materials
requires the measurement of the dynamic stress intensity
factor, K I D , for materials failing in a predominantly
cleavage mode with a relatively small plastic zone size.
Previous determinations of KIDin opaque materials have
utilized several optical techniques including reflection
photoelasticity (1)-(3)t and caustics (4x6).Recently,
methods based on strain gauge techniques (3),(7x9)
have received attention. These methods are attractive
from an experimental point of view owing to the relative
simplicity involved in the measurement. However, the
strain response at the gauge location as the crack passes
near this point must be defined in sufficient detail to
extract K I D from the strain-time trace.
The method to be presented here is an extension of the
static methodologies originally introduced in (10) and
further advanced in (11) and (12). A series solution for a
crack propagating at constant velocity is utilized to
describe the strain at a point. By proper orientation of
the gauge, the effect of the constant coxstress is eliminated. However, the importance of retaining the next
higher order parameter (rl/’ term) is shown.
k
The MS.ofthis paper was received at the Institution on 19 June 1989 and accepted
for publication on 18 January 1990.
t Rpferences are given in the Appendix.
0 IMechE
+
(24
V)
where c, and c2 are the longitudinal and shear wave
speeds, respectively. The complex valued functions Z,,
2, , Yl and Y, are defined as
J
Zk=
C
Ajz’,-”’;
k = 1, 2
(44
j=O
M
Yk=
A series solution for the stresses near a crack propagating at constant velocity (13)(14), is converted in (9)to the
strain sensed by a gauge oriented at an angle a as shown
in Fig. 1. The strain in the direction of the gauge is
2 p g = /ll[{k(l: - 1;) + (1 + 1:) cos 2a)
x Re 2, -8, cos 2a Re 2,
+ (k(1: - 1:) + (1 + 1:) cos 2a)
x Re Y, - /13 cos 2a Re Y,
+ 21, sin 2a( -1m 2 , + Im 2, - Im Y,)
+ /14 sin 2a Im Y,]
(1)
- v)/(l
and p, v are the shear modulus and Poisson’s ratio,
respectively. The velocity-dependent functions A, and 1,
are defined as
2 DYNAMIC STRAIN FIELD
JOURNAL OF STRAIN ANALYSIS VOL 25 NO 3 1990
= (1
1
B,Z;;
k = 1,2
(4b)
m=O
with the velocity transformed zkdefined in Fig. 2 as
Truncating the series expressions given by equation (4)
after the third term, gives the strain sensed by the gauge
as a three-parameter model
2 P * = A , fo
+ A I f 1 + Bo go
177
1990
0309-3247/90/07O0-0177
(6)
$02.00 + .05
Downloaded from sdj.sagepub.com at PENNSYLVANIA STATE UNIV on May 11, 2016
R. J. SANFORD, J. W. DALY AND J. R. BERGER
a for various Poisson’s ratios are given in (10). For the
steel used in this investigation v = 0.30 and either
a = 61.3 degrees or a = 118.7 degrees satisfies equation
(8). It is important at this point to examine which of the
two choices for a gives strain data most suitable for the
accurate determination of KID.
To examine the two possible choices for a which
satisfy equation (8) it is helpful to consider a modified
form of equation (9) given by
2P&$Ao = f o + (AI/AO)fl
(10)
Since the position, y,, of the gauge above the crack propagation path is fixed, it is possible to eliminate r from
equation (7a) and (7b) by noting
r = y$sin 8
(1la)
This substitution leaves 8 as the only unknown position
coordinate. The relation
Fig. 1. Definition of strain gauge orientation angle, a
where
x = y$tan 8
fo = Blr;1/2[cos(01/2){k(1:- 1;) + (1 + 1:)
x cos 2aj
+ 21, sin (8,/2)sin 2a]
+ 21, sin (8,/2) sin 2a
f l = Blr:’2[Cos(8,/2){k(1;- 1
:) + (1 + 1:)
x cos 2a
x cos 2a) - 21, sin (8,/2)sin 2a]
x cos 2a - 2A1’sin (8,/2) sin 2a
go
= B l ( l : - 1;Xk
+ cos 2a)
The go function in equation (7c) can be made identically
zero by selecting the orientation angle, a, as
cos 2a = - k
Then equation (6) becomes
(8)
(9)
2Pg = A0 fo + A l f l
and the B, parameter is eliminated as a variable with no
loss in generality of the three-parameter model. Values of
(11b)
indicates that x and 0 are interchangeable and that the
modified strain 2 p $ A O given in equation (10) can be
shown as a function of either x or 8 as the crack passes
beneath the gauge position. A family of curves are
obtained as shown in Figs 3 and 4 for selected values of
AlIAO.
Consider first the response of the gauge when
a = 118.7 degrees. This choice for a was used in (3)based
on the argument that the peak in strain will coincide with
the gauge position. However, as shown in Fig. 3, the
peak in 2 p J A O occurs at the gauge position (x = 0), if
and only if, A , = 0. For other values of A,/Ao the crack
tip extends beyond the gauge before the peak in cg is
recorded. This fact implies that recording the time of the
peak strain is not sufficient to accurately locate the crack
tip position. Examination of Fig. 3 also shows that the
maximum value of the modified strain 2,u&$A0 depends
strongly ( x 30 per cent) on the ratio A l / A o. It is clear
then that a one parameter analysis with the gauge oriented at a = 118.7 degrees is inherently prone to error.
Finally, it should be noted that a zero crossing does not
occur in Fig. 3 with the gauge at this orientation.
Orienting the gauge at a = 61.3 degrees results in the
gauge response shown in Fig. 4 with features which
improve the accuracy in the K I D determination. First,
-8
30 20
-20
-40
-60
-80
Position. X (mm)
--b
C
Fig. 2. Multiple coordinate systems used in the dynamic analysis
178
0
Fig. 3. Modified strain as a function of position for a = 118.7 degrees
(y, = 10.5 mm and c = 656 m/s)
JOURNAL OF STRAIN ANALYSIS VOL 25 NO 3 1990
Downloaded from sdj.sagepub.com at PENNSYLVANIA STATE UNIV on May 11, 2016
6 IMechE
1990
AN IMPROVED STRAIN GAUGE METHOD FOR MEASURING K,,
( I ) Time -peak to peak
(2) Time-maximum peak to zero
-2I
1
I
1
30
20
10
!
0
I
I
-20
1
I
I
-40
I
1
-60
q,T0)
I
-80
Position. X (mm)
Fig. 4. Modified strain as a function of position for a
( y , = 10.5 mm and c = 656 m/s)
=
61.3 degrees
Time -zero to minimum peak
top
and most important, is that the peak value of 2p&$A, =
6.40 f 0.19 m - l for a wide range of A , / A , . This fact
implies that K I D can be determined from the peak of the
strain-time trace within f3 per cent without knowledge
of A, or the position of the crack tip. A second difference
is the zero crossing of E , which occurs as the crack
extends beyond the gauge. This is a well-defined crossing
which can be used as an additional datum for locating
the position of the crack tip as described in (9). For this
investigation A ] / A 0 will be determined from characteristic features of the Eg-t (or E ~ - x curves.
)
(4) Time at 314 of maximum
amplitude
3 FEATURE EXTRACTION
As shown qualitatively in Fig. 5, there are several
approaches for characterizing the temporal response of
the gauge as the crack passes near its position. Of the five
possibilities shown, three of the cases involve quantities
taken from either the negative peak, the minimum strain
or the time of occurrence of the minimum strain.
However, the negative peak is so shallow that the
resolution of either x or t is poor when interpreting the
experimental strain-time records. For this reason any
approach involving the negative peak is not particularly
useful in terms of characterizing the strain response.
There remain two possibilities of the five cases shown
in Fig. 5 for analysing the strain response. The time from
the positive peak to the zero crossing (case 2) is a strong
characteristic due to the sharpness of the positive peak
and the well-defined zero crossing. However, this quantity is not as sensitive to A , / A , as case 4 where the time is
measured between the rise and fall of strain at 3/4 of the
peak strain,
This quantity is quite sensitive to
A , / A , , is readily obtained from the experimental
records, and is used to establish A ] / A o for this investigation. The fraction of peak strain used is, in itself, not
significant; however, the value of 3/4 was selected in
order to insure that moderately large values of strain
would be used but still retaining enough separation
between the curves for various values of A J A , to be
easily approximated.
From the numerically generated data shown in Fig. 4,
(At)3l, can be extracted for several A , / A , values to
obtain the relation between these two quantities shown
in Fig. 6. Unfortunately, the crack speed must be known
a priori in order to generate Fig. 4 and determine (At)3,4.
JOURNAL OF STRAIN ANALYSIS VOL 25 NO 3 1990
0 lMechE
Fig. 5. Characteristic features of the strain-time records for a
degrees in steel
=
61.3
However, as indicated in (8), the dynamic effect is small
at crack velocities normally encountered in engineering
materials under essentially constant loading. Therefore,
little error can be expected in utilizing the curve shown in
Fig. 6 to determine A , / A , for a wide range of crack
velocities.
With A , / A , known a single master curve, similar to
one of those shown in Fig. 4, can be generated. It is then
a simple matter to extract A, and hence K I D from the
2
8
1
0
Ratio, A , I A , , ( m - ' )
Fig. 6. Time parameter (At)3,4 versus coeficient ratio A J A ,
1990
Downloaded from sdj.sagepub.com at PENNSYLVANIA STATE UNIV on May 11, 2016
119
R. J. SANFORD, J. W. DALY AND J. R. BERGER
strain gauge trace knowing 2 p $ A 0 at a selected point.
This process is detailed in the next section.
4 EXPERIMENTAL ANALYSIS
An experiment was performed to demonstrate the procedure described above. The specimen was fabricated
from 4340 alloy steel heat treated to a hardness of R , =
51. The over-sized compact tension Specimen, shown in
Fig. 7, was fabricated and saw-cut to simulate the initial
crack and a blunted chevron notch was filed at the tip to
control the load at initiation. Shallow side grooves equal
to 5 per cent of the specimen thickness were used to
control the crack propagation path.
The specimen was instrumented with six strain gauges
each with an active grid length of 3.18 mm (0.125 in).
Following the argument presented in (lo), errors due to
strain gradient over the gauge length were negligible. The
gauges were oriented at a = 61.3 degrees and were
placed along a gauge line with y, = 10.5 mm (0.413 in) on
12.7 mm (0.50 in) centres. The first gauge was located at
a/W = 0.53. The gauges were connected to high performance bridge and amplifier units with a frequency
response of DC to 120 kHz. The strain gauges were calibrated using the shunt calibration method immediately
prior to the experiment. Digital oscilloscopes were then
used to record the output signals using a sampling rate of
200 ns per point.
A closed-loop servo-hydraulic test system was used to
load the specimen. Voltagetime traces were downloaded from the oscilloscope bubble memories onto a
personal computer. A commercial spread sheet program
was then used to post process the data. The resulting &,-t
traces for the six gauges referenced to a common time
base are shown in Fig. 8. Excellent agreement was
obtained between the experimental traces and the predicted behaviour shown in Fig. 4.
The time of the peak strain was plotted as a function of
the gauge position x as shown in Fig. 9. The x-t relation
is essentially linear except for a slight deviation associated with the reading from gauge No. 1. The corresponding velocity was considered constant at 656 m/s (25 800
in/s) over the recording interval.
Next, the time measurement (At)3l4 needed for determining the ratio AJA, was made from the strain-time
-60
-20
20
60
140
100
180
Time (ps)
Fig. 8. Strain-time records for the compact tension specimen
traces shown in Fig. 8. The results showing
for
each signal are given in Table 1.
The results were very consistent and permitted an
average value of
to be taken as 21.3 ps. Using this
average value in Fig. 6 gives A JA, = - 23.6 m - I .
The single master curve for AJA, = -23.6 m-' and
c = 656 m/s which corresponds to the experiment conducted here is shown in Fig. 10. The most direct determination of A, can be made from the peak value of
2p$A0 = 6.53. Then A, = 2p$6.53 and K I D is
= 24(2~)&$6.53
(12)
Since side grooves were employed in this study, equation (12) must be modified to account for their presence.
The side groove correction factor is given by
K,D
c, = ( B / B , ) ' / ~
(13)
C = 656 rnls
1-15
Surface groove
both sides
-20
0
20
40
60
i0
100
Time to peak, rp (ps)
Fig. 9. Crack tip versus time used to obtain crack velocity
Table 1. (A&,* for the six strain gauge traces
0.500 thick
(12.7)
I IFig. 7. Specimen geometry and strain gauge locations,inches (mm)
180
Gauge
(At)n,4 ols)
1
2
3
4
5
6
21.4
21.0
21.6
21.0
21.8
21.0
JOURNAL OF STRAIN ANALYSIS VOL 25 NO 3 1990 0 IMechE 1990
Downloaded from sdj.sagepub.com at PENNSYLVANIA STATE UNIV on May 11, 2016
AN IMPROVED STRAIN GAUGE METHOD FOR MEASURING KID
A,IA,
Dynamic strain response
= -23.6 m - '
C = 656 mls
1.28
/
-80
1.24 -
-60
1.20 -
-40
1.12 -
-20
1.16
0
-
AI/AO
1.04-
(m-')
1.08
1 .OO
-2!
-30
1
!
,
-10
,
,
30
,
10
,
,
,
50
0
I
,
0.1
70
0.2
0.3
0.4
0.5
(
Velocity, c/cR
Position, X (mm)
Fig. 10. Dynamic master curve for A J A , = -23.6 m - ' (y, = 10.5
m m and c = 656 m/s)
where B is the gross specimen thickness and B, is the
reduced thickness through the side grooved region. For
the specimen used in this investigation, B = 12.7 mm
(0.500 in), B, = 11.6 mm (0.455 in) and C, = 1.048. The
Table 2. Peak strains and K , , for each gauge
Gauge
Peak strain @m/m)
KID(MPaJm)
KID(ksiJin)
1
2404
2036
1923
1881
1884
1867
153.8
130.3
123.0
120.3
118.0
119.4
140.1
118.6
112.0
109.6
107.4
108.8
2
3
4
5
6
Table 3. Strains and KIDat lops before peak
Gauge
Strain (pm/m)
KID(MPaJm)
KID(ksiJin)
1
2010
1663
1589
1547
1523
1501
157.2
130.1
124.3
121.0
119.1
114.4
143.2
118.6
113.3
110.2
108.6
106.9
2
3
4
5
6
200
I80
working equation for K I D is then
The peak strains were taken from the oscilloscope
records shown in Fig. 8 to give E ~ Results
.
for K I D are
presented in Table 2.
In Table 2 K I D decreases during propagation from
gauge 1 to gauge 2. This is due to both the change in
velocity and a decrease in the stress intensity factor following initiation from an elevated K , . From gauge 2 to
gauge 6, the crack propagates at essentially constant
KID.In principle any point on the strain-time record can
be used to determine K I D by selecting the corresponding
point on the master curve. For example, instead of utilizing the peak value in 2p&/AOfrom the master curve
shown in Fig. 10, consider the value of 2p$A0 that
occurs 10 ps prior to the peak strain. From Fig. 10, this
value is 2p$A0 = 5.34 and equation (14) then becomes
where the strain E, is measured 10 ps before the peak
strain on the strain-time traces. Results of this analysis
are presented in Table 3. A comparison of these results
shows a very close correspondence between the two sets
of determinations. The results determined 10 ps before
the peak strains shown in Table 3 are within k2.2 per
cent of the measurements made at the peak (Table 2).
Using the values in both tables as well as additional data
for different times prior to the peak, if desired, allows us
to construct a graph of K I D as a function of position
along the crack line, as illustrated in Fig. 11. As discussed
previously, K I D decreased during the initial phase of propagation and then remained essentially constant over the
remaining period of measurement. The results K I D determined with the dynamic strain-position relations were
consistently 2 per cent higher than those obtained using
the static strain-position equations.
Comparison of the results presented here using the
dynamic strain versus position equations with the results
previously presented (10) using the static strain versus
position equations shows that the dynamic results are
3
h
E
hE
i,
4
Fig. 12. Velocity correction factor, Cv versus normalized velocity,
c/cR(cR= Rayleigh wave speed)
60
0
.*,
Peakdata
- 10 ps data
,
,
,
,
,
I
,
0
60
80
100
Position, X (mm)
Fig. 1 1 . Dynamic stress intensity factor, K,,,as a function of position
along the crack line
0
20
I
40
JOURNAL OF STRAIN ANALYSIS VOL 25 NO 3 1990
0 IMechE
1990
Downloaded from sdj.sagepub.com at PENNSYLVANIA STATE UNIV on May 11, 2016
181
R. J. SANFORD. J. W. DALY AND J. R. BERGER
consistently 2 per cent higher than the static results. This
elevation in K I D due to dynamic influences is consistent
with the predictions of reference (10). As a consequence,
an alternative analysis procedure would be to use the
static strain versus position equation for the analysis;
that is
2,u&$A0 = r-’/’{k cos (e/2)
6 CONCLUSIONS
-(1/2) sin 8 sin (38/2) cos 2a
+(1/2) sin 8 cos (38/2) sin 2a)
+(A1/Ao)r”’ cos (e/2)
x
the &,-time record are employed to determine K,,, the
error due to uncertainty in position will be less than 3 per
cent. Combining all of the sources of error, the r.m.s.
error can be expected to be about + 6 per cent in 68 per
cent of the K I D determinations.
{k + sin’ (e/2) cos 2a - (1/2) sin e sin 2af
(16)
and correct the results using the velocity correction
factor shown in Fig. 12. The graphs of Fig. 12 are for
y, = 10.5 mm; however, the velocity correction is not a
strong function of gauge height and the curves shown
can be used for any reasonable gauge height and normalized velocity up to c/cR= 0.3 with less than 4 per cent
error in C , .
A method has been described to measure K I D for running
crack using strain measurements. By considering various
orientations of the gauge relative to the crack path, the
strain gauge response was optimized for the KIDdetermination. It was demonstrated that, of the two possible
choices for gauge orientation angle which satisfy equation (8), the acute angle is superior. With this choice of
angle even moderate errors in estimates of A , / A o have
insignificant influences on the final result for realistic
crack velocities. The method was applied to the measurement of the dynamic fracture toughness of a hard and
brittle 4340 alloy steel. The results for KID determined
with the dynamic strain-position relations were consistently 2 per cent higher than those obtained using the
static strain-position equations.
5 ERROR ANALYSIS
The errors involved in the determination of K I D with
strain gauges are due to four sources which include:
elastic constants, instrumentation, the truncated-series
strain representation and the gauge position and/or
orientation relative to the crack tip. The elastic constants, ,u and k, appear either directly or indirectly in the
K I D - & , relations and errors in these constants are carried
over to the calculation of K I D . The dynamic strains are
measured with wide-band amplifiers and high sampling
rate digital converters to ensure signal fidelity. The sensitivity of the overall instrumentation system including
power supply, bridge circuit, AD converter, and oscilloscope is determined with shunt calibration using 0.1 per
cent precision resistors. Even with these techniques it is
generally accepted that the anticipated error in dynamic
strain gauge measurements is of the order of 3-5 per cent
and this error carries over directly to the KIDcalculation.
The KID-&,relation used in this analysis is a three term
approximation to the exact strain state around a propagating crack. The truncation of the series introduces an
error in the calculation which depends on the instantaneous distance of the gauge from the crack tip as the crack
passes below the gauge. In this study only a small
portion of the total recorded strain data was used. Only
data collected over region 10.5 mm < I < 18 mm and 40
degrees < 0 < 70 degrees ahead of the crack was used in
the analysis. The error in the three term representation of
the strain field is estimated to be less than 3 per cent
within this zone of measurement.
The remaining source of error is due to positioning
and orienting the gauge. The orientation angle can be
held to f 1.5 degrees by exercising care during installation. The position of the gauge relative to the crack tip
at any instant of time is determined by locating the peak
in the strain-time record and relating the time of
occurrence of the peak with the position through Fig. 10.
As can be seen from Fig. 4, the location and magnitude of
the peak strain is not sensitive to the A, term and the
effect of positioning errors is minimal. If peak values of
182
ACKNOWLEDGEMENTS
The authors would like to thank the National Science
Foundation for support under grant no. MSM-85-13037,
Naval Air Development Center under contract No.
N62269-87-C-0288 and Oak Ridge National Laboratory
under contract No. 19X-07778C.
APPENDIX
REFERENCES
(1) KOBAYASHI, T. and DALLY, J. W. ‘Dynamic photoelastic
determination of the a-K relation for 4340 alloy steel’, Crack
arrest methodology and applications, (edited by G . T. Hahn and M.
F. Kanninen), A S T M S T P 711,1980,189-210.
(2) DER, V. K., BARKER, D. B., and HOLLOWAY, D. C. ‘A split
birefringent coating technique to determine dynamic stress intensity factors’, Mechs Res. Comm., 1978 5,313-318.
(3) SHUKLA, A,, AGARWAL, R. K., and NIGAM, H. ‘Dynamic
fracturestudies on 7075-T6 aluminum and 4340 steel using strain
gauges and photoelastic coatings’, Engng Fracture Mech., 1988,
13,501-515.
(4) SHOCKEY, D., KALTHOFF, J. F., KLEMM, W., and
WINKLER, S. ‘Simultaneous measurements of stress intensity for
fast running cracks in steel’, Expl Mech., 1983,23,14&145.
(5) RAVI-CHANDAR, K. and KNAUSS, W. G . ‘An experimental
investigation into dynamic fracture I : crack initiation and arrest’,
lnt. J . Fracture, 1984,25247-262.
(6) ROSAKIS, A. J. ‘Analysis of the optical method of caustics for
dynamic crack propagation’, Engng Fracture Mech., 1980, 13,
331-347.
(7) READ, D. T. ‘Analysis of strains measured during a wide plate
crack arrest test’, Proceedings of the 20th Session of the German
Society for Materials Testing, 1988, Working Group on Fracture,
Frankfurt.
(8) DALLY, J. W. and SANFORD, R. J. ‘On measuring the instantaneous stress intensity factor for propagating cracks’, Proceedings,
International Conjerence on Fracture V l I , 1989, Pergamon Press,
Oxford, pp. 3222-3230.
(9) BERGER, J. R., DALLY, J. W., and SANFORD, R. J. ‘Determining the dynamic stress intensity factor with strain gauges using a
crack tip locating algorithm’, Engng Fracture Mech., in press.
(10) DALLY, J. W. and SANFORD, R. J. ‘Strain gauge methods for
measuring the opening mode stress intensity factor, K,’, Expl
Mech., 1987,27,381-388.
JOURNAL OF STRAIN ANALYSIS VOL 25 N O 3 1990
Downloaded from sdj.sagepub.com at PENNSYLVANIA STATE UNIV on May 11, 2016
0 IMechE
1990
AN IMPROVED STRAIN GAUGE METHOD FOR MEASURING K,,
(11) BERGER, J. R. and DALLY, J. W. ‘An overdeterministic
approach for measuring K,using strain gauges’, Expl Mech., 1988,
28,142-145.
(12) DALLY, J. W. and BERGER, J. R. ‘Determining K ,and K,,in a
mixed mode stress field using strain gauges’, Proceedings I986
SEM Spring Conference on Experimental Mechanics, 1986, Society
for Experimental Mechanics, Bethel, CT, pp. 603-612.
JOURNAL OF STRAIN ANALYSIS VOL 25 N O 3 1990
0 IMechE
(13) IRWIN, G. R. ‘Constant speed semi-infinite tensile crack opened
by a line force, P, at a distance, b, from the leading edge’, Lehigh
University Lecture Notes on Fracture Mechanics, 1968.
(14) CHONA, R. The stressfield surrounding the tip of a crack propagating in afinite body, PhD thesis, University of Maryland, 1987.
1990
Downloaded from sdj.sagepub.com at PENNSYLVANIA STATE UNIV on May 11, 2016
183