Transactions on Engineering Sciences vol 13, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
Further studies on T* integral under plane stress
crack growth
Y. Omori/H. Okada^L.Ma/S.N. Atluri,* A.S. Kobayashi/R Tan'
"Department of Mechanical Engineering, University of Washington,
^Computational Mechanics Center, Georgia Institute of Technology,
^Technical Center, Federal Aviation Administration, Atlantic City
International Airport, New Jersey 08405, USA
Abstract
T*£-integral values associated with stable crack growth in thin 2024-T3
aluminum single edge notched (SEN) and central notched (CN) specimens were
computed directly from the crack tip displacement field obtained by moire
interferometry. These results agreed with the T*e obtained from an elasticplastic finite element (FE) simulation of the stable crack growth in a SEN
specimen. These T*e variations with stable crack growth also agreed with those
determined experimentally in the CN specimens thus suggesting that for a given
F£, T*e can be used as a stable crack growth and a ductile fracture criteria.
1
Introduction
In a series of papers, May and Kobayashi [1] and Tran et al [2] showed that
under stable crack growth, the ASTM J-resistance curve does not represent the
crack tip state of stress since J is not a path independent integral and rapidly
approach zero with the decrease in the size of the integration contour. Brust et
al [3,4], on the other hand, showed through finite element (FE) analysis, that
the T* integral [5] remained a steady state value under loading, unloading and
under stable crack growth. The J and the T* integrals are contour integrals,
which are identical in form, where the former and latter are based on
deformation and incremental theories of plasticity, respectively. The T* integral
is thus the summation of incremental AT* determined during the plastic
deformation process. In the presence of crack growth, the J-integral contour,
which is constant in shape, moves together with the crack tip while the T*integral contour maintains its original geometry but elongates with the extending
Transactions on Engineering Sciences vol 13, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
512 Localized Damage
crack tip. J and T* integrals are both path dependent in the presence of
unloading. Although the near field J value has been shown to vanish [2], the
corresponding near-field T* value remained constant [3,4] during stable crack
growth. The latter is more appealing physically and thus a series of
experimental and numerical studies on the T* integral was initiated through
coordinated efforts at the University of Washington and the Georgia Institute of
Technology.
In this paper, we present T*-integral values which are determined
directly from the measured displacement field surrounding stably growing
cracks in thin 2024-T3 aluminum, single edge-notched (SEN) and centralnotched (CN) specimens.
2 Method of approach
Moire interferometry was used to determine the orthogonal displacement fields
surrounding stably growing cracks in thin 2024-T3 aluminum alloy, SEN and
CN specimens and T*^, which refers to a T* integral value of a small contour,
F£, was computed. FE analysis was also conducted on truncated FE models of
the SEN and CN specimens where the measured displacements, which varies
with stable crack growth, were prescribed on the truncated boundaries and the
T*^ was computed.
3 Experimental procedure
A new moire interferometer [6], which combines the advantages of geometric
moire and traditional moire, was used to measure the large strains in the vicinity
of the crack tip. The method uses a low-spatial frequency, i.e. 40 lines per
mm, steep grating on a mirror-finished specimen surface to construct high
contrast moire fringes on the specimen surface. Figure 1 shows the 2024-T3
SEN and CN specimens used in this study.
Okada and Atluri [7] have shown through numerical experimentation
that for a contour adjacent to the crack, the trailing portion of the contour
integral can be ignored since the surface traction acting on such near-crack
contour vanishes. Without the contour integration involving the unloaded
region, the contour integral ahead of the crack tip can be evaluated by using the
deformation theory of plasticity for estimating the state of stress from the
measured displacement field. The experimentally impractical procedure of
incrementally evaluating AT* was thus avoided and T* was determined directly
using the measured displacement field surrounding a partial contour near and in
front of the crack tip.
Transactions on Engineering Sciences vol 13, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
Localized Damage 513
4 Numerical procedure
The finite element model consisted of one half of a truncated SEN specimen and
one quarter of a truncated CN specimen. The measured displacements were
prescribed on the truncated horizontal boundary, also shown in Figure 1, and
an elastic-plastic FE analysis was conducted with the time varying edge
boundary condition and the crack tip location. The two FE models consisted of
graded finite elements with the smallest element of 0.25 mm distributed along
the crack. Again, incremental evaluation of AT* was avoided as Pyo et al {8]
has shown that the T* computed by using the current states of stress and strain
based on incremental theory of plasticity was practically equal to the sum of
AT*. T*^ value was computed for the same F^ used in the experimental
analysis as well as for near-crack contours, F^, of different sizes. Numerically
these contour integrals were evaluated by an equivalent domain integral [9] for
increased numerical accuracy.
5 Results
Figure 2 shows typical moire interferometry patterns corresponding to the uand v-displacement fields of a CN specimen and the F^ contour of e = 2 mm
from the crack. 6 = 2 mm, which is sufficiently removed from the slanted crack
front where the state of plane stress prevailed, was used for F
Figure 3 shows that the experimentally and numerically determined T*
continued to increase with crack extension in the SEN specimen, possibly due
to the large specimen rotation and hence the lack of constraint in this specimen.
The FE analysis also showed that while T*^ varied with F^ it was independent
of the size of the outer contour for a given F
Figure 4 shows the experimentally and numerically determined T*e for
the CN specimen. Unlike the SEN specimen, T*e appears to reach a steady
value expected of stable crack growth.
Thickness: 0.8
.8
'
-*—— so.e *»
\
n V
714| 4
f 25,4
o ^.
~ f
-C
Thickness: 0.8
M
I
U
•f -f
44
m
•wf
*— 50,8-w
Figure 1: SEN and CN 2024-T3 aluminum specimens.
114.3
»
t
t f 50.8
44\
Transactions on Engineering Sciences vol 13, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
514 Localized Damage
Figures 5 and 6 shows that the experimentally and numerically
determined crack tip opening displacements (CTOA) for the SEN and CN
specimens, respectively. The experimentally and numerically determined
Figure 2: u- and v-displacement moire fringe patterns of CN specimen.
180.0
160.0
140.0 -
1 120.0
CO
S; 100.0 2
c?
|f
80.0 -
^"
40.0
#
n
A
v
0
60.0
20.0
0.0 0.0
1.0
Experimental T* for SEN
FEA T* for SEN: h = 2.5 mm
FEA T* for SEN: h = 3.0 mm
FEA T* for SEN: h = 3.5 mm
FEA T* for SEN: h = 4.0 mm
T
2.0
Aa
3.0
(mm)
4.0
5.0
Figure 3: T*g variations with stable crack growth in SEN specimen.
Transactions on Engineering Sciences vol 13, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
Localized Damage 515
CTOA are in excellent agreement and are characterized by an initial maxima
which was not observed in the T*^ variation with crack extension. While
Dawice et al [9] attributed this maxima to tunneling of the crack front, this
maxima was also duplicated by Tran et al who used plane-stress FE analysis to
replicate the CTOA variation.
CONCLUSIONS
1. Experimental and numerical procedures for determining T*^ and CTOA are
presented.
2. Variations in T*^ and CTOA with stable crack growth in 2024-T3 aluminum
SEN and CN specimens were determined experimentally and numerically
and were found to be in reasonable agreement with each other.
3. T*^ reaches a steady state value with crack extension and is suitable for
characterizing stable crack growth.
#
^
160.0 -
1
•
_ /g-.g-.g...^...
b &@#
•
_ < eft«"*
1
140.0 -
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4»
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*
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-
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Experi mental T*
FEA T* : h = 2.5 mm
FEA T* : h = 3.0 mm
FEAT* : h = 3.5mm
FEA T* : h = 4.0 mm
1
0.0
0.0
1.0
2.0
3.0
Aa (mm)
4.0
Figure 4: T*e variation with stable crack growth in CN specimen.
5.0
Transactions on Engineering Sciences vol 13, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
516 Localized Damage
ACKNOWLEDGMENT
The results reported here were obtained through a FAA Research Grant 92G-005.
REFERENCES
1 . May, G.B. and Kobayashi, A.S. Plane stress stable crack growth Integral/
HRR field, Int. J. Solid and Structures , 37, No. 617, 857-881, 1995.
2. Tran, D. K., May, G. B. And Kobayashi, A. S. Further studies on JIntegral under plane stress crack growth, submitted for publication, 1996.
3. Brust, F.W., Nishioka, T., Atluri, S.N. and Nakagaki, M. Further studies
on elastic-plastic stable fracture utilizing the T* integral, Engineering
cj., 22, 1079-1103, 1985.
4. Brust, F.W., McGowan, JJ. and Atluri, S.N. A combined numerical/
experimental study of ductile crack growth after a large unloading, using
T*, J and CTOA criteria, Engineering Fracture Mechanics, 23,.537-550,
1986.
25.0
20.0 -?-
O FEA
# Moire
• Measured
d)15.0
mo
o
5.0 -
0.0
0.0
2.0
4.0
6.0
Aa (mm)
8.0
T
10.0
12.0
Figure 5: CTOA variations with stable crack growth in SEN specimen.
Transactions on Engineering Sciences vol 13, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
Localized Damage 517
5. Stonesifer, R.C. and Atluri, S.N. On a study of the (AT) and C* integrals
for fracture analysis under non-steady creep, Engineering Fracture
Mechanics, 16, 769-782, 1982.
6. Wang, F.X., May, G.B. and Kobayashi, A.S. Low-spatial-frequency
steep grating for use in moire interferometry, Optical Engineering, 33,
1125-1131, 1994.
7. Okada, H. and Atluri, S.N., T*^ integral evaluation from experimental
displacement field for a plate with stably propagating crack: development of
calculation procedure and implications of T*^, to be submitted, 1996.
8. Pyo, C.-R., Okada, H. and Atluri, S.N. An Elastic-plastic finite element
alternating method for analyzing wide spread fatigue damage in aircraft
structures, Computational Mechanics., 16, 62-68, 1995.
9. Dawicke, D.S., Sutton, M., Newman, J.C. and Bigelow, C.A.
Measurement and analysis of critical CTOA for thin-sheet aluminum alloy
materials, Fracture Mechanics 25th Symposium, ASTM STP 1220, eds. F.
Erdogan and R. J. Hartranft, 358-379, 1995.
30.0 25.0 -
1
)
S? 200
[0,
0
O
o FEA
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• Measured
o
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15.0 " 0
•
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. .£
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• £
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00
0.0
1.0
1
2.0
3.0
4.0
#....#*
•
1
5.0
6.0
Aa (mm)
Figure 6. CTOA variation with stable crack growth in CN specimen.