Mechanics of Materials 28 Ž1998. 147–154
T´) integral as a crack growth criterion
Yoshika Omori
a,)
, Albert S. Kobayashi a,1, Hiroshi Okada
Paul W. Tan d,4
b,2
, Satya N. Atluri
c,3
,
a
d
Dept. of Mechanical Engineering, UniÕersity of Washington, Box 352600, Seattle, WA 98195-2600, USA
b
Dept. of Mechanical Engineering, Kagoshima UniÕersity, Kagoshima 890, Japan
c
Computational Mechanics Center, Georgia Institute of Technology, Atlanta, GA 30332-0356, USA
Federal AÕiation Administration, William J. Hughes Technical Center, Atlantic City Int’l. Airport, Atlantic City, NJ 08405, USA
Received 8 January 1997; revised 14 August 1997
Abstract
T´)-integral values associated with stable crack growth in thin 2024-T3 aluminum single edge notched ŽSEN., central
notched ŽCN. and compact ŽCT. specimens were evaluated experimentally and numerically along a near-crack contour, G´ ,
which elongated with crack extension. The surface displacement fields, which were determined by Moire´ interferometry and
by an elastic–plastic finite element ŽFE. simulation of the measured stable crack growth, were used to determine the T´).
The experimentally and numerically determined T´) ’s and those of the crack tip opening angles ŽCTOA., were in excellent
agreements with each other. Variations in T´) and CTOA with crack extension of all three specimens were in essential
agreement, albeit the continually increasing T´) of the SEN specimen. q 1998 Elsevier Science Ltd. All rightd reserved.
Keywords: Crack; Integral values; Surface
1. Introduction
For the past 7 years, the authors and their colleagues have used experimentally determined displacement fields to compute directly the J-integral in
thin aluminum fracture specimens ŽMay and
Kobayashi, 1995.. The contour integration was performed using the definition of the J-integral ŽRice
and Rosengren, 1968. with an added assumption that
)
Corresponding author.
E-mail: ask@u.washington.edu.
2
E-mail: okada@mech.eng.kagoshima-u.ac.jp.
3
E-mail: satya.atluri@cad.gatech.edu.
4
E-mail: tanp@admin.tc.faa.gov.
1
the elastic–plastic response of the specimen material
could be represented by a power hardening law.
These J-integral values agreed well with the corresponding elastic values under low load and with the
known elastic–plastic solutions ŽKumar et al., 1981.
at higher loads prior to stable crack growth. Under a
small stable crack growth of 1 to 3 mm, however,
the near-field J-integral values increasingly deviated
from the known solution and were as much as one
quarter of that computed by Kumar et al. Ž1981.. For
J evaluated along a 2-mm square contour surrounding the stably extending crack tip, the J value was
approximately one tenth of the ASTM Ž1987. far-field
J and was in qualitative agreement with the numerical experiment conducted by Brocks and Yuan
0167-6636r98r$19.00 q 1998 Elsevier Science Ltd. All rights reserved.
PII: S 0 1 6 7 - 6 6 3 6 Ž 9 7 . 0 0 0 5 9 - 8
148
Y. Omori et al.r Mechanics of Materials 28 (1998) 147–154
Ž1989.. The computed and measured HRR displacements ŽHutchinson, 1968. perpendicular to the crack,
i.e., the Õ-displacements, agreed well but substantial
differences were found in the corresponding values
for the displacements parallel to the crack, i.e., the
u-displacements.
Attempts were also made to characterize the second order term in the crack tip asymptotic displacement field which is defined here as the difference
between the measured and HRR displacements. The
experimental results described above showed that
neither the two parameter characterization by the
Q-stress, nor the T-stress, prevailed in the presence
of stable crack growth. Thus, without the presence of
an HRR field or an HRR field with the higher order
terms at the crack tip, the much heralded ASTM
J-integral ŽASTM, 1987. loses its physical significance as the strength of the HRR singular field. In
fact, Turner Ž1990. noted that the cumulative quantity of the ASTM J, which rises with an increasing
crack length, is not a true crack driving force or the
resistance to crack extension. Despite these misgivings, the J–R concept continues to dominate the
fracture community.
As an alternative to the ASTM J, the T´) ŽStonesifer and Atluri, 1982., which is also a path dependent integral based on incremental theory of plasticity, circumvents much of the difficulties encountered
by the J-integral as shown by Brust et al. Ž1985,
1986. through numerical simulation of stable crack
growth experiments. More recently, the authors and
their colleagues ŽOmori et al., 1996a,b. have used
experimental, numerical and hybrid experimentalrnumerical techniques to demonstrate the utility
of the T´) integral in characterizing stable and rapid
crack growth in aluminum fracture specimens. The
results of this feasibility study are summarized in the
following sections.
nated as T´) . The T´) fracture parameter as defined
by Stonesifer and Atluri Ž1982. is:
T´) s
HG
´
ž
Wn1 y t i
E ui
E x1
/
dG
Ž 1.
where t i are the surface tractions on the contour G´ ,
W is the strain energy density and n1 is the first
component of the normal to the curve. G´ is an
arbitrary small contour immediately surrounding the
crack tip and more importantly it elongates together
with crack extension as shown in Fig. 1. T´) , as
defined by Eq. Ž1., is identical in form as the J-integral and therefore T´) coincides with J where the
deformation theory of plasticity prevails. In terms of
incremental theory of plasticity, T´) integral at the
end of the Nth load steps is the sum of DT´) which
is the incremental change of T´) over a load step or:
N
T´) s Ý DT´)
Ž 2a .
where
DT s
HG
´
DWn1 y Ž t i q D t i .
ED ui
E x1
y D ti
E ui
E x1
dG
Ž 2b .
The current T´) , as defined by Eq. Ž2a., is thus
dependent on the prior loading history, a property
that is essential for elastic–plastic analysis under
2. T ) integral
Extensive numerical analyses by Brust et al. Ž1985,
1986. has shown that the T ) integral in the very
vicinity of the crack tip reaches a plateau, unlike the
local J-integral which vanishes with crack growth
ŽBrocks and Yuan, 1989., with crack extension under creep and cyclic loading. This local T ) is desig-
Fig. 1. Contours for contour integration and equivalent domain
integral ŽEDI..
Y. Omori et al.r Mechanics of Materials 28 (1998) 147–154
crack growth. Although such incremental analysis
can be routinely conducted by finite element ŽFE.
analysis, it is not practical in experimental analysis
as the cumulative experimental errors per load step
will eventually overwhelm the sought data. Fortunately, Pyo et al. Ž1995. have shown, through numerical analysis, that the total T´) integral computed
directly by using the stresses and strains based on the
incremental theory of plasticity, was for practical
purpose equal to the summed DT´) of Eq. Ž2a..
Thus, Eq. Ž1. can be used for crack growth study
without the cumbersome incremental procedure provided the states of stress and strain are based on the
incremental theory of plasticity.
Since the stress field near a crack tip cannot be
calculated accurately due to the inherent large stress
and strain gradients, the ‘equivalent domain integral
ŽEDI.’ method ŽNikishkov and Atluri, 1987. was
used to calculate T´) from remote values as:
T´) s
H
G´
ž
H
Wn1 y t i
sy
Ay A ´
ž
E ui
E xi
/
Sd A
E
E x1
E
Ž WS . y
E xi
ž
St i
E ui
E x1
//
=d Ž A y A´ .
where S s 1 on G´ and S s 0 on a remote closed
path, G , and V´ is the area between G and G´ .
The magnitude of ´ is governed by a characteristic dimension which assures that a plane stress state
exists along the integration contour of G´ . This
distance, ´ , is equated to the plate thickness after the
work of Narasimhan and Rosakis Ž1990.. 5 For a
plane strain state, this characteristic distance could
be several times the crack tip radius.
Since the inception of the T ) integral concept,
much of the sporadic papers on this subject were
limited to numerical analysis which were then verified with far field parameters, namely the load vs.
load-line displacements ŽBrust et al., 1985, 1986..
5
The three-dimensional elastic–plastic FE analysis of a flat
crack in plate by Narasimhan and Rosakis Ž1990. showed that the
plane stress state prevailed at one half of the plate thickness from
the crack tip. Since a 100% shear lip is the norm in ductile
fracture, the minimum distance, ´ , was conservatively picked as
the plate thickness.
149
T ) was also determined indirectly through a hybrid
experimental–numerical analysis.
3. Method of approach
3.1. Experimental procedure
The experimental procedure consisted of measuring the two orthogonal displacement fields surrounding a stably growing crack in the single edge notched
ŽSEN., the central notched ŽCN. and the compact
ŽCT. specimens using Moire´ interferometry. Fig. 2
shows the three 2024-T3 aluminum specimens which
are discussed in this study. t in the figure refers to
the specimen thickness. A coarse cross diffraction
gratings of 40 linesrmm was used to determine the
large plastic strains surrounding the extending crack.
A special Moire´ interferometry procedure ŽWang et
al., 1994., which combines the advantages of geometric and traditional Moire´ interferometry and uses
a low frequency Moire´ grating for measuring large
strains in the vicinity of the crack tip, was developed
for this ductile fracture study. A special four-beam
Moire´ interferometry bench was also constructed for
use with this low frequency Moire´ grating.
In order to circumvent the inaccuracy inherent in
crack tip analysis, T´) was computed by using a
partial equivalent EDI encompassing the frontal region of the crack tip. The strains and stresses along
the contour, G´ , for computing T´) integral were
determined directly from the orthogonal displacement fields obtained by Moire´ interferometry. The
stresses corresponding to the total strains were then
computed using the equivalent stress–strain and the
measured uniaxial stress–strain data of the 2024-T3
sheet. This use of the deformation theory of plasticity to compute stresses does not account for the
unloading process which occurs in the trailing wake
of the extending crack. However, by restricting the
integration contour very close to and along the extending crack, Okada and Atluri Ž1996. have shown
that the contour integration trailing the crack tip can
be neglected by virtue of the closeness of the integration path, G´ , to the traction free crack surface. This
approximation not only simplifies the integration
process since the contour integral is restricted to
Y. Omori et al.r Mechanics of Materials 28 (1998) 147–154
150
Fig. 2. 2024-T3 aluminum fracture specimen configuration. Moire´ grating placed on shaded area.
frontal portion of G´ , but also eliminates the undesirable effects of the deformation theory of plasticity
which must be used to compute the stresses in the
unloaded portion of G´ .
The crack tip opening angle ŽCTOA. was also
computed by the angle subtended by the measured
crack opening displacement ŽCOD. at a distance 1
mm from the crack tip.
3.2. Numerical procedure
men truncated at a 10-mm distance from the crack.
While the entire SEN specimen could have been
easily modeled with this two-dimensional FE analysis, the truncated FE model allowed more detailed
analysis around the crack tip with a limited number
of FEs. In contrast, the entire CT specimen, albeit
half due to geometric symmetry as shown in Fig. 3,
was modeled in the plane stress FE analysis because
of the awkward location of the loading pin. The
measured orthogonal u- and Õ-displacements along
the truncated width of the SEN and CN specimens
The plane stress, FE models of the SEN and CN
specimens represented only a portion of the speci-
Fig. 3. FE mesh configuration for CT specimen.
Fig. 4. Stress–strain relationship of 2024-T3 aluminum alloy.
Y. Omori et al.r Mechanics of Materials 28 (1998) 147–154
151
Fig. 5. Moire´ fringe patterns Ž D a s 1 mm.. 2024-T3 CT specimen.
and the measured pin displacement of the CT specimen together with the measured instantaneous crack
length. were used to drive the FE model in its
generation mode ŽKobayashi, 1979.. The FE analysis
was based on the incremental theory of plasticity
using the measured equivalent stress-strain relation
shown in Fig. 4.
The T´) integral along an elongated contour surrounding the stably growing crack of the fracture
specimens was then computed. Unlike the stresses
used in the experimental procedure for T´) evaluation, the FE analysis provided stresses which accounted for the unloading effect in the trailing wake
of the extending crack tip. Therefore, the entire
contour was used for T´) evaluation where numerical errors in the FE data in the vicinity of the crack
tip were masked by the EDI ŽNikishkov and Atluri,
1987.. To recapitulate, the T´) evaluation procedures
for the Moire´ and FE studies differ in that the former
involved only the frontal segment of a near-field
contour, G´ as shown schematically in Fig. 5, while
the latter involved an EDI over the entire crack
length also shown schematically in Fig. 1.
mm ahead of the extended crack. A Moire´ grating of
75 linesrmm was used in this particular test. All
other experiments with SEN, CN and CT specimens
were conducted with a Moire´ grating of 40 linesrmm
as described previously.
Fig. 6 shows the load vs. crack extension in the
2024-T3 CT specimen. Since the measured pin displacement was prescribed in this generation analysis
ŽKobayashi, 1979. of the FE studies, the match
between the computed and measured loads indicates
the accuracy in the FE modeling of stable crack
growth. However, the match of the applied load,
which is applied far from crack tip, alone does not
guarantee good match in the crack tip field parameters.
4. Results
Fig. 5 shows typical Moire´ interferometry patterns
of the u- and Õ-displacement fields of an 2024-T3
CT specimen and the partial contour with ´ s 3.0
Fig. 6. Load variation during crack extension. 2024-T3 aluminum
alloy CT specimen.
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Y. Omori et al.r Mechanics of Materials 28 (1998) 147–154
Fig. 7 shows excellent agreement between the
T´) , which were determined experimentally and by
FE analyses, for the CT specimen. h in the figure
refers to the outer contour distance, as shown in Fig.
1, for the EDI computation. While ´ was close to
the plate thickness of 3.1 mm, h varied from 3.5 to
4.5 mm in the FE analysis. The FE-determined T´)
remained essentially invariant with h as expected.
Also shown is the computed J-integral for ranging
from 0.5 mm to 9.0 mm as well as the ASTM
J-integral ŽASTM, 1987.. As mentioned in Section
1, the far-field J-integral has no resemblance of the
near-field J-integral which nearly vanishes with
crack extension. The far field J, i.e., ´ s 7.0 and 9.0
mm, agrees well with the T´) values.
Fig. 8 shows the T´) , which were determined
experimentally and by FE analyses, for the SEN, CN
and CT specimens. T´) ’s in the CN and CT specimens reached the same steady state value of about
120 MPa-mm while T´) of the SEN specimen continue to increase to about 140 MPa-mm. The increasing T´) is possibly due to the lack of constraint in
the SEN specimen caused by the continuous grip
rotation with crack growth.
Fig. 9 shows the CTOA variations, which were
determined by Moire´ interferometry and FE analysis,
Fig. 7. T´) and J-integral variations during crack extension.
2024-T3 aluminum alloy CT specimen. t s 3.1 mm.
Fig. 8. Experimentally and FEA determined T´) integral values.
2024-T3 aluminum SEN, CN and CT specimens.
with crack extension in the CT specimen. Although
the T´) results for the CT specimen was obtained by
plane stress FE analysis, the CTOA for the CT
specimen was computed by plane strain FE analysis
in view of the thickness of the CT specimen, i.e.,
t s 3.1 mm. The experimental and FE computed
CTOA’s are in good agreement throughout much of
the stable crack growth.
Fig. 10 shows the CTOA variation in the three
2024-T3 SEN, CN and CT specimens determined by
Moire´ interferometry. Despite the differences in geometry and thickness, all CTOA’s reached a constant
Fig. 9. CTOA during crack extension. 2024-T3 aluminum alloy
CT specimen.
Y. Omori et al.r Mechanics of Materials 28 (1998) 147–154
153
can categorically be considered a material property.
A point of reference is the immense and futile
ASTM efforts to establish J-integral as a crack
growth parameter.
Acknowledgements
Fig. 10. Experimentally determined CTOA. 2024-T3 aluminum
SEN, CN and CT specimens.
58. This steady state value also coincides with that of
Dawicke et al. Ž1995. who tested 2.3 mm thick
2024-T3 aluminum CT and CN specimens.
5. Conclusions
Ž1. T´) computed from the displacement fields
obtained experimentally and from FE analysis were
in good agreement with each other.
Ž2. The FE-determined T´) values, when evaluated using the EDI method, are independent of the
size of the outer contour, h, and converge to a
stationary value for a smaller inner contour, G´ where
´ is equal to the thickness of the specimen.
Ž3. The steady state T´) value for the three specimens were essentially the same and suggests that the
steady state T´) could be used to characterize stable
crack growth in thin aluminum sheets.
Ž4. The computed and measured CTOA reached a
steady state value of 58 after a stable growth of about
D a s 2.0 mm in all 2024-T3 specimens regardless of
specimen geometry and thickness.
6. Discussion
Although the results of this feasibility study indicates that the T´) integral could be used to characterize stable crack growth, similar study must conducted with different specimen configurations of
varying dimensions as well as materials before T´)
The results reported here were obtained through
FAA Grant 92-G-005 and DOE Grants DE-FG0694ER14490 and DE-FG0694ER 14491. The authors
wish to acknowledge the contributions of Messrs. P.
Lam, L. Ma, K. Perry and J. S. Epstein.
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