Crack Linkup: An Experimental Analysis
by L. Ma, A. S. Kobayashi, S. N. Atluri, and P.W. Tan
ABSTRACT--The Ts* integral was used to assess stable crack
growth and crack linkup in 0.8 mm thick 2024-T3 aluminum
tension specimens with multiple site damage (MSD) under
monotonic and cyclic loads. The T~* values were obtained
directly from the recorded moir6 fringes on the fracture specimens with and without MSD. The T~* resistance curves of
these fracture specimens of different geometries were in excellent agreement with each other. The results suggest that
T~* is a material parameter which can be used to characterize crack growth and linkup in the absence of large overloading. T~* based crack growth and net-section-yield based crack
linkup criteria for MSD specimens are proposed. The crack tip
opening angle (CTOA) criterion can also be used to correlate
crack growth larger than 2 mm.
KEY WORDS--Fracture mechanics, moir6 interferometry,
stable crack growth, crack linkup, multiple site damage (MSD)
Introduction
The failure scenario of multiple site damage (MSD) is
governed by a long lead crack which is formed by successive
linkups of in-line short cracks. Gruber et al.1 used linear
elastic fracture mechanics (LEFM) to predict the MSD crack
linkup and fracture load for stiffened panels and compared
his prediction with the test results. Swift 2 postulated crack
linkup when the two plastic zones, as per Irwin's formula, of
two adjacent cracks connected. Nishimura 3 used a strip yield
model to estimate the coalescence condition of the plastic
zones for multiple cracks in a riveted stiffened sheet. When
compared with limited experimental results, neither LEFM
nor the LEFM based plastic zone always predicted a MSD
crack linkup.
The purpose of this study is to establish experimentally
a methodology based on a newly developed ductile fracture
criterion, i.e., the T~* integral, to predict stable crack growth
in a ductile material and crack linkup of a structure containing
MSD.
T~ Integral Criterion
The T~* integral, 4'5 which is a near-field integral based on
the incremental theory of plasticity, varies with the near field
L. Ma is a Doctor, United Technology Research Center, Structural Integrity
Group, East Hartford, CT06108. A. S. Kobayashi is a Professor Emeritus,
University of Washington, Department of Mechanical Engineering, Seattle,
WA 98195-2600. S. AT.Atluri is a Professor, University of California, Los
Angeles', Center for Aerospace Research and Education, Los Angeles, CA,
90095-1597. R W. Tan is a Doctor, FAD<William J. Hughes Technical Center,
Atlantic City International Airport, NJ 08405.
Original manuscript submitted: August 7, 2000.
Final manuscript received." December 17, 2001.
integration contour, F. For a straight crack which is subjected
to a self-similar straight crack propagation
r~ = f
[Wnk
-
sijnjui,k] d F ,
(1)
where nk, Sij and ui are the direction normal, stress and displacement, respectively, W is the strain energy density and
i, j, k = 1, 2. The indices 1 and 2 represent the Cartesian
coordinates parallel and perpendicular to the straight crack.
Since the direction normal, n2 = 0, along that portion of I'~
parallel to the crack, and the stresses, s12 and s22, vanish when
I~ is very close to the traction-free crack, the integrand in
eq (1) vanishes in the trailing wake of the crack. 6 This nearness, ~, for an integration contour, I~, close to a traction-free
crack is set to the thickness of the specimen in order to guarantee a state of plane stress along the integration contour. 7
By using the stress/strain fields generated by the incremental theory of plasticity, Pyo et al.8 showed, through numerical
experiments, that T~*can be computed without summing A T~*
for each incremental crack extension. As a result, not only is
the experimentally impractical procedure of evaluating A T~*
avoided but T~*can be determined directly from the measured
displacement field surrounding a partial contour in front of
the crack tip.
CTOA Criterion
The critical crack tip opening angle (CTOA) criterion assumes that stable crack growth occurs when the CTOA made
by a point on the upper surface of a crack, the crack tip, and
a point on the lower surface reaches a critical value. For convenience, a point 1 mm behind the crack tip has been used
to define the CTOA. Extensive experimental results for thin
aluminum fracture specimens have shown that, after an initial transient period, the CTOA remains constant throughout
Mode I stable crack growth. 9' 10
Method of Approach
The experimental procedure consisted of measuring the
two orthogonal displacement fields surrounding a stably extending crack in an aluminum specimen using moir6 interferometry with a co~se ~{oss diffraction grating of 40 lines/ram.
Single-edge noi~h: (SEN) and center notch (CN) specimens
mach)~n~d~from A1 2024-T3 clad aluminum sheet of thickness 0.8 mm were used to establish a'tearing resistance curve
of the material based on the T~* integral. The test Specimens
consisted of CN specimens of the same material with multiple
site damage (MSD). Starter cracks were fatigued at a stress
ratio of R = 0.1 and C~max= 0.3Cry, where C~maxand Cyy are
Experimental Mechanics
,, 147
the maximum and yield stresses, respectively. CN specimens
with starter cracks, which were saw cut and sharpened with
a razor blade, were also tested. All cracks were oriented in
the L-T direction. Special buckling guides were designed to
prevent out-of-plane buckling of the fracture specimen due to
the inherent compressive s22 stress along the crack. Unlike
traditional buckling guides, these guides were spaced apart
for moir6 interferometry analysis of the translating crack tip
region. Without buckling guides, the out-of-plane deformation of the center region was as much as 10 mm. The results of the SEN and CN specimens and MSD2 specimens
with two cracks, MSD3 specimens with a center/lead crack
approaching two holes with MSD cracks and MSD5 specimens with a center/lead crack and four holes with MSD
cracks are reported in this paper. Figure 1 shows the SEN
and CN specimens used to generate the T~* resistance curve.
The length and width of the CN specimen varied from W =
50.8, 152.4, 254.0 mm and L = 228.6, 317.5 and 431,8 mm.
Figure 2 shows the MSD2, MSD3 and MSD5 specimen
configurations.
The test section of all specimens was covered with a
crossed moir6 diffraction grating of 40 lines/mm over a region
of 50 mm x 100 mm. This coarse moir6 grating was necessary due to the large scale plastic yielding, which resulted in
moir6 fringe patterns too dense to resolve by the traditional
high density grating. A master cross grating was transferred
by contact printing on to a relatively thick photo-resist coating
on a highly polished specimen surface. The resulting deep
cross-grooved photoresist coating and the exposed, highly
reflective specimen surface constituted a diffraction grating
which withstood the large straining in the crack tip region.
The specimen was illuminated by a four-beam moir6 interferometer for sequential recording of the two moir6 interferometry fringe patterns. The fringe patterns corresponded to
two orthogonal u- and v-displacement fields which, unlike in
traditional moir6 interferometry, could be viewed from any
angular orientation.11
The u- and v-displacements corresponding to the moir6
fringes were input to interpolation software which computed
the total strains along the integration contour. The stresses
corresponding to the total strains were then computed using
the equivalent stress-strain relation deduced from the measured uniaxial stress-strain data of the A1 2024-T3 sheet.
This use of the deformation theory of plasticity to compute
stresses did not account for the unloading process, which occurred in the trailing wake of the extending crack. Since the
contour integration of T~* trailing the crack tip can be neglected by virtue of the closeness of the integration path, F~,
to the traction free crack surface, this unloading process did
not enter in the T~* evaluation. This partial contour integral
was evaluated along the open square in the very vicinity of
the moving crack tip, i.e., ~ = 1 mm, following the procedure
described by Okada and Atluri. 6 The partial contour integral
also eliminated the use of the deformation theory of plasticity for computing the stresses from the total strains in the
unloaded region trailing the extending crack tip.
The crack tip opening angle (CTOA) was obtained from
the angle at the crack tip and between the measured crack
opening displacement (COD) 1 mm away.
T~ Results
SEN and CN Specimens
Thirteen SEN and ten CN with fatigued or saw-cut crack
tips were used to establish the basic material tearing resistance. Due to the ductile nature of A1 2024-T3, the crack tip
exhibited a large amount of crack tip blunting prior to stable crack growth. The initial crack tip blunting eliminated
any difference in the stable crack growth responses of fracture specimens with saw cut and fatigue cracks. The residual
strengths of the 152.4 and 254 mm wide specimens without
buckling restraint were 7% and 15% lower than those with
buckling restraint. T~* integral data of one SEN and five CN
specimens with different widths, together with three 3-mm
thick compact (CT) specimens from a previous investigation 7
are shown in Fig. 3. The higher T~* curve for the 50.8 mm
wide CN specimen was attributed to the specimen size effect.
Crack growth in the 50.8 mm wide specimen did not initiate
until the net section stress reached 1.2 times the yield stress,
whereas it started at approximately 0.8C~y in the 152.4 and
254 mm wide CN specimens. Other than the 50.8 mm wide
CN specimens, the T~* data of the SEN, CN and CT specimens in Fig. 3 coincide, thus suggesting that T~*is a geometry
independent material property provided the net section stress
in the fracture specimen is below the yield stress. The T~*
resistance curve represents an average of this data.
MSD2 Specimen
Fig. 1--SEN and CN specimens
148 9 VoL 42, No. 2, June2002
The MSD2 specimen contained two equal-length cracks
extending toward each other. The initial fatigue cracks were
either 2.2, 3.2, 4.2 or 4.9 mm in length. Buckling guides
were not needed for these narrow specimens. Figure 4 shows
typical u- and v-moir6 fringe patterns and the increase in
measured T~* values with stable crack growth. While the
T~* values correlated well with the T~* resistance curve, the
MSD2 specimens were only able to sustain a stable crack
growth of 1.5 mm on each adjoining crack before T~*reached
(a)
(c)
(b)
Fig. 2--MSD2, MSD3 and MSD5 specimens. (a) MSD2, (b) MSD3, (c) MSD5
gardless of the specimen configuration and that crack linkup
occurred when T~* reached the plateau of the T~* resistance
curve.
160
gg_
140
................................................................................................
120
...........
loo
....m
8o
O
:
: []
t.m
..........
'~....*i, ~
~'
mlU
,&
~ ...............................
7
.
.
.
.
@
::
.
.
.
.
.
.
.
.
.
.
.
.
60
.
.
.
.
.
.................
.
.
.
.
.
.
.
.
.
.
.
.
.
~
.
.
.
.
.
.
0 ......
.
.
.
.
.
.
MSD5 Specimens
.
. . . . . .
~7~V
_~
.
:ii
.~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
9
m
•
9
v
SEN (W=25,4mm) 1
CN 0N=50.8mm)
CN (W=152,4mm)
CN 0N=254rnm)
CT(W=100mm)
40
...........................................................
20
............... "................ ". . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .:................ ................
SEN & CN spei~irnen thickness = 0.8rnm
QT specimen tNckness = 3.0 mm
0
6
8
10
12
Crack Extension (mm)
Fig. 3--SEN, CN and CT T8* resistance curves
the plateau of the T~* resistance curve in Fig. 3. Stable crack
growth started when the net section stress reached 1.2C~y and
remained at this level throughout the short crack growth in the
MSD2 specimens. In contrast, the CN specimens had stable
crack growth of about 3 to 14 mm for net section stresses of
1.2cry to 0.8Oy, respectively.
MSD3 Specimen
The center lead crack in the MSD3 specimens was about
50.8 mm in length and the MSD cracks were 0.3 to 5.2 mm in
length. The center crack extended at approximately the same
stress level of 260 MPa (0.87Cry) and unstable fracture occurred at a net section stress of 410 to 425 MPa. The strength
reduction was equal to the net sectional area reduction due to
the MSD cracks, thus indicating that the maximum load was
reached at crack linkup. Figure 5 shows typical moir6 fringe
patterns and T~* variations. A comparison of Figs. 3 and 5
shows that all cracks propagated at the same T~* value re-
Specimens MSD5_08 (ligament = 12.7 mm and aMSD =
2.5 mm) and MSD5_13 (ligament 20.32 mm and aMSO =
2.5 mm) failed simultaneously with the first cracklinkup. The
experimental T~* versus crack extension for these specimens
are shown in Figs. 6(b) and 7(a). In both cases, all cracks
exhibited stable crack growth at T~* values related to the T~*
resistance curve. Crack tips C and D, as identified in Fig. 6(a),
extended approximately 3 mm and 1.5 mm prior to linkup.
Crack linkup occurred when T~* reached the plateau portion
of the T~* resistance curve and the net section stress at linkup
were at/below the yield stress.
Specimen MSD5_16, which had the same crack configuration as MSD5_08, was subjected to three cyclic loads, as
shown in Fig. 7(b). At each cycle, the load was increased
until crack tip D extended approximately 0.8 mm. The specimen was then unloaded to 10% of the maximum load. Experimental T~ was evaluated for each crack extension as the
specimen was cyclically loaded. Figure 7(b) shows that Ts*
values under cyclic and monotonic loads are the same during
crack growth. Thus the T~* resistance curve, which was generated from a monotonically loaded CN specimen, may also
be used to characterize low cycle fatigue crack growth.
CTOA Results
The experimentally determined CTOA obtained from the
SEN and CN specimens correlated reasonably well. All
CTOAs scattered during the initial stage (Aa ~ 1-2 ram) of
crack tunneling and shear lip formation, as shown in Fig. 8.
The CTOA then settled to a constant 5 to 6 degrees throughout
the remaining crack growth process.
The large scatter in CTOA during the crack tunneling and
blunting stage indicated that the CTOA is not an appropriate
ExperimentalMechanics
9
149
(a)
(b)
160
160
i
140
140 ~- .......... ............9.................................
o
120
.......... IS;
~'1oo
............. ........... i...........
~'too
o~ei
E
~- 80
120 -- ........... i----. .o .....................................
..........1 ~ ........................................
E
- 80
o
............................................ :...........
t-- 60 .......................................................
................................
4O
O
|
B
20
s =1ram:
i~
0
CN
Crack B
Crack C
o
9
9
.................................
CN
Crack B
Crack C
20 ................................... 9. . . . . . ..........
9
Craci linkup
I
0
40 J
~=lm~
i~
i
0
I
0
2
3
4
Crack Extension (ram)
C r a c k linkup
L
3
4
Crack Extension (mm)
(c)
(d)
Fig. 4 - - T ~ o f M S D 2 _ 2 a n d M S D 2 _ 8 s p e c i m e n s .
c o r r e s p o n d i n g t o v - d i s p l a c e m e n t field, ( c ) M S D 2
(a) M o i r 6 f r i n g e s c o r r e s p o n d i n g
2, a n d (d) M S D 2 8
(a)
to u-displacement
(b)
160
160
.......... 4........... , ........ , ........... , ..........
140
140
:
E
E
a_
120
100
..........
i
9
!'""
"0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i9
O
~
60
" 9
,2
............................................
40. : ....... :i........... ~
i
20
13_
84
i
.......... i b o ' ~
i
......................
, con,e,c,aok
,MSOCrack
ON - CN08
r=lm~
b Cra~k linku~
:
o
9
9
..............
im
80
60
#
40
i
i ..........
i
9 o
"E
(
.m..-..m...:...o. ............................ !...........
g~
mo::
~
~- 12o
E
100
..... o.--.i~......... i.......... ~...........i...........
80
E
field, (b) M o i r e f r i n g e s
o
20
!
ii]
IF =lmrr),
MSD Crack
CN - ON08
i i:> Cra~klinkup
0
0
1
3
4
Crack Extension (ram)
(c)
3
4
"rack Extension (mm)
5
(d)
Fig. 5 - - T ~ f o r M S D 3 _ 0 9 a n d M S D 3 _ 1 3 s p e c i m e n s . (a) M o i r e f r i n g e s c o r r e s p o n d i n g t o u - d i s p l a c e m e n t
c o r r e s p o n d i n g t o v - d i s p l a c e m e n t field, ( c ) S p e c i m e n M S D 3 _ 0 9 , a n d (d) S p e c i m e n M S D 3 _ 1 3
150 9
Vol. 4 2 , No. 2, J u n e 2 0 0 2
field, (b) M o i r e f r i n g e s
160
140
:
120
i~ i
P
!
ioi
~
~-Ioo
E
o-
-o-
-o-
D
C
80
-o
B
A
60
40
20
Crack C
C N - CN10
o
.......................
"
cr ok ,oko
2
3
4
5
Crack Extension (mm)
(a)
(b)
Fig. 6--T~ for MSD5_08 specimen. (a) Crack configuration, and (b) MSD5 08 Lig = 12.7 mm, amsd = 2.5 mm
160
160 -
140
140
120
120
~hoo
. . . . . . . . . . . p . - . o--. ..~ * i
E
.................
g'loo
.......... f;,,.2
........ p.....s ........
E
8o
8o
60
, m
6o
.O...
I
9
....................
Crack D
Crack C
Crack B
C N - CNIO
9
40
9
@
2 0 "
!
1
0
@ack linkut
p
2
3
Crack Extension (mm)
:
:
~i Crack linkup
0 4-,-~, , i . . . . i . . . . i . . . . " . . . . ' . . . .
0
1
2
3
4
5
Crack Extension (mm)
(a)
(b)
Fig. 7--T~* for MSD5_13 and MSD5 16 specimens. (a) MSD5_13 Lig = 20.3 mm, aresd
12.7 mm, amsd = 2.5 mm, low cycle fatigue
16"
:"
:.
.
:.
.
;
; ..........
:
:. . . . . . . . . . . . . . . . . . . .
A
lO
<
o
I,-
......
i
i
! ..........
i .....................
8 ~ ...... 9
4~e
~
6:
. . . . .
4
. . . . . . . . . . .
2 . -
0
,
,
,
..........
.
SEN
CN (W=50.8mm)
CN (W=50.8mm)
CN (W=254mrn)
..........
; ..................
"
,<~
A
9
@
O
.
.
..........
: ........
; ..........
;
i ..........
i ......
: . . . . . . . . . .
:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
:
:
;
2
,
,
,
;
4
.
.
.
.
While the utility of a T~* resistance curve in assessing
stable crack growth in MSD specimens was demonstrated, no
new crack linkup criterion emerged from this experimental
investigation. Figure 9 shows that crack linkup will occur
in all but the MSD specimens with the shortest remaining
ligament of 7.62 mm. The reduction in net section ligament
with stable crack growth had triggered a plastic collapse. The
merger of two adjoining large strain concentration fields in
the shortest remaining ligament of 7.62 mm triggered plastic
collapse prior to net section yield.
,.
.
.,. ~ .........
.
.
.
6
,
8
,
,
,
,
10
,
,
2.5 mm, and (b) MSD5_16 Lig =
Discussion
]
14
12 . . . . 9 . . . . .
=
,
12
Crack Extension (mm)
Fig. 8--Experimental CTOA of SEN and CN resistance curves
crack criterion for short crack growth of less than 2 mm. The
T~* criterion, on the other hand, can cover the entire crack
growth spectrum since T~* rises gradually from the initial
crack growth stage and reach a constant value during stable
crack growth.
Conclusions
An experimental study of the T~*integral fracture criterion
and its use in predicting crack link up in tension panels with
MSD have been presented. The study included five different
crack configurations, ranging from one to five cracks, to study
the Te* integral in a single crack environment as well as in
a multiple cracks environment with crack interaction. The
following conclusions were gleaned from this study.
1. Stable crack growth in MSD panels can be correlated
with the T~* resistance curve of the material.
Experimental Mechanics
9
151
1 A0
0.9
9
Ligament=7.62mrn (MSD5)
9
[]
9
A
Ligament=12.7mm (MSD5)
Ligament=12.7mm (MSD3)
Ligament=20.3mm (MSD5)
Ligament=19-23mm (MSD3)
References
i
,/~
~,,,,~
.... : : ........
a
03
:~0.8
i
i
9
i
i
9
Z
a
co
0.7
o_
0.6
0.5
....
0.5
i . . . . . . . . . . . . . . . .
0.6
0.7
0.8
0.9
1.0
Amsd/ANon-MSD
Fig. 9--Strength reduction vs. area loss due to MSD
2. Except for the initial phase of stable crack growth, i.e.,
Aa ~ 2 mm, the CTOA resistance curve can be used
to correlate stable crack growth.
3. Crack linkup occurs when net ligament length between
the lead and MSD cracks reaches a critical length for
plastic collapse.
Acknowledgments
This work was supported by Federal Aviation Administration Grant 9 I-G-0005.
152 9
Vol. 42, No. 2, June 2002
1. Grube~ M.L., Wilkins, K.E., and Worden, R.E., "Investigation of Fuselage Structure Subjected to Widespread Fatigue Damage," FAA-NASA Symposium for Continued Airworthiness of Aircraft Structures, DOT/FAA/AR97/2,2, 439-460 (1997).
2. Swift, T., "Widespread Fatigue Damage Monitoring--Issues and Concerns," FAA/NASA International Symposium on Advanced Structural Integrity Methods for Airframe Durability and Damage Tolerance, NASA CP
3274, 829-870 (1994).
3. Nishimura, T., "Strip Yield Analysis of Plastic Zones on Coalescence
of Plastic Zones for Multiple Cracks in Riveted Stiffened Sheet," ASME J. of
Engineering Materials and Technology, 121, 352-359 (1999).
4. Stonesifer, R.C. and Atluri, S.N., "'On a Study of the (TE) and C*
Integrals for Fracture Analysis Under Non-steady Creep," Eng. Fracture
Mechanics, 16, 769-782 (1982).
5. Atluri, S.N., Nishioka, T., and Nakagaki, M., "Incremental PathIndependent Integrals in Inelastic and Dynamic Fracture Mechanics," Eng.
Fracture Mechanics, 20 (2), 209-244 (1984).
6. Okada, H. and Atluri, S.N., "Further Studies on the Characteristics
of the T~ Integral: Plane Stress Stable Crack Propagation in Ductile Materials," Computational Mechanics, 23, 339-352 (1999).
7. Omori, Y., Kobayashi, A.S., Okada, H., Atluri, S.N., and Tan, P.,
"T~ as a Crack Growth Criterion," Mechanics and Materials, 28, 147-154
(1998).
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 Structure," Computational Mechanics, 16, 62-68 (1995).
9. Dawicke, D.S., Sutton, M.A., Newman, J.C. Jr., and Bigelow, C.A.,
"Measurement and Analysis of Critical CTOAfor an Aluminum Alloy Sheet,"
Fracture Mechanics, 25th Volume, ed. F. Erdogan. ASTM STP 1220, 358379 (1995).
10. Dawicke, D.S., Plascik, R.S., and Newman, J.C.J., "Prediction of
Stable Tearing and Fracture of a 2000-Series Aluminum Alloy Plate Using
a CTOA Criterion," Fatigue and Fracture Mechanics: 27th Volume, eds.
R.S. Plascik, J. C. Newman Jr., and N.E. Dowling, ASTM STP 1296, 90-104
(1997).
11. Wang, FX., May, G.B., and Kobayashi, A.S., "Low-spatial Frequency
Steep Geometric Grating for Use in Moiri Interferometry," Optical Engineering, 33 (4), 1125-1131 (1994).