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Fatigue & Fracture Of Engineering Materials & Structures, 33, 12, pp. 822-831,
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Neutron diffraction measurements of residual stresses around a crack
tip developed under variable-amplitude fatigue loadings
Lee, S. Y.; Rogge, R. B.; Choo, H.; Liaw, P. K.
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doi: 10.1111/j.1460-2695.2010.01490.x
Neutron diffraction measurements of residual stresses around a crack
tip developed under variable-amplitude fatigue loadings
S. Y. LEE
1
, R. B. ROGGE
2
,H.CHOO
1
andP.K.LIAW
1
1
Department of Materials Science and Engineering, The University of Tennessee, Knoxville, TN 37996, USA,
2
Canadian Neutron Beam Centre,
National Research Council Canada, Chalk River, Ontario, Canada K0J 1J0
Received in final form 19 April 2010
ABSTRACT
The spatially resolved neutron-diffraction residual stress mappings were performed on
five compact-tension (CT) specimens subjected to various variable-amplitude fatigue
loadings (e.g. overload, underload and their mixed loads) during fatigue crack propaga-
tion. Three principal residual-stress components (i.e. longitudinal, transverse and normal
stresses) were measured as a function of the distance from the crack tip along the crack-
propagation direction. The shape of respective crack tips on the five CT specimens was
examined using scanning electron microscope. The results show the distinct residual-
stress fields near the crack tip and significant changes in the crack-tip geometry for five
different loading cases. It is thought that the combined effects of the changes in the
residual-stress state and crack-tip geometry seem to be a key factor to account for the
observed transient crack-growth phenomena.
Keywords crack tip; fatigue crack propagation; neutron diffraction; residual stress;
variable-amplitude fatigue.
INTRODUCTION
In the case of numerous fatigue-critical structure compo-
nents, fatigue crack propagation under service conditions
generally involves random or variable-amplitude loadings
rather than constant-amplitude loading.
1
The accurate
understanding and control for the crack resistance of ma-
terials subjected to variable-amplitude loadings, for ex-
ample, overload and/or underload, are therefore crucial
to develop the damage tolerance design and lifetime pre-
diction methodology.
Residual stresses are one of the contributory factors to
failure in structural components. Withers
2
demonstrated
that when unexpected failure occurs it is often because
residual stresses have combined critically with the ap-
plied stresses, or because they, together with the pres-
ence of unknown defects or poor microstructures, have
dangerously lowered the applied stresses at which fail-
ure will occur. Residual stresses also play a significant
role in the fatigue crack growth behaviour. It is gener-
ally known that compressive residual stresses are found
to decrease the crack propagation rates, whereas tensile
Correspondence: P. K. Liaw. E-mail: pliaw@utk.edu
residual stresses yield the opposite effect.
3
More specifi-
cally, in terms of the crack-growth retardation phenom-
ena following a single tensile overload, many researchers
reported that the enlarged compressive residual stresses
after a tensile overload are one of the possible retardation
mechanisms, slowing down the crack-growth rates in the
retardation period.
4–7
Makabe et al.
8
demonstrated that
the tensile residual stresses developed by a compressive
underload are an important consequence of the reversed
plastic flow, leading to the reduction of crack-opening
level and acceleration of crack-growth rate.
Various models depending on the residual stresses
have also been developed to predict the fatigue crack
growth behaviour under constant-amplitude or variable-
amplitude loadings.
9,10
However, Lam et al.
11
pointed
out that the models predicting the residual-stress effect
on fatigue crack growth have not been completely quan-
tified, due to a task of difficulty to accurately measure
the residual-stress distribution. Thus, the direct residual-
stress measurements near the crack tip influenced by
prior plastic deformation will be of importance to the
improvement of a fatigue lifetime prediction model, as
well as a better understanding of the crack propagation
behaviour.
822
c
2010 Blackwell Publishing Ltd.
Fatigue Fract Engng Mater Struct
33, 822–831
Fatigue & Fracture of
Engineering Materials & Structures
NEUTRON DIFFRACTION MEASUREMENTS OF RESIDUAL STRESSES 823
Nondestructive diffraction methods (e.g. high-energy
synchrotron X-ray diffraction or neutron diffraction) are a
powerful technique in the direct measurement of internal
strains/stresses in the bulk sample.
12–20
Steuwer et al.
15
investigated the imaging of fatigue cracks and associated
crack-tip strain field using synchrotron X-ray diffraction
and tomography. They observed a significant compres-
sive zone at and behind the crack tip following a 100%
overload. Croft et al.
16
examined the local strain fields
in the vicinity of fatigue-crack tips during in situ loading
using synchrotron X-ray diffraction. They found a large
compressive residual strain near the crack tip immediately
after the overload, but there was no difference between
the strain change (ε
yy
) curves before and immediately
after the overload. They also reported the transfer of load
response between the overload position and the propa-
gated crack tip following the overload. More recently,
Lee et al.
20
showed the development of internal strains
around a crack tip during tensile overloading, compres-
sive underloading, and their combinations using neutron
diffraction.
In this investigation, the direct measurements of
residual-stress distribution are carried out as a function of
the distance from the crack tip using neutron diffraction,
immediately after applying the same loading conditions
as our previous study
20
(i.e. a tensile overload, a compres-
sive underload and their mixed loads during fatigue crack
growth). The shape of respective crack tips for the differ-
ent loading cases was examined using scanning electron
microscope. The results will be useful for the develop-
ment of more accurate residual-stress-based prediction
models, as well as the computational simulations.
EXPERIMENTAL DETAILS
The fatigue crack growth experiments were conducted
on a nickel-based HASTELLOY C-2000 (56%Ni–
23%Cr–16%Mo, in weight percent)
21
compact-tension
specimen (Fig. 1a) prepared according to the American
Society for Testing and Materials (ASTM) Standards
E647-99.
22
This material has a single-phase face-centered
cubic (FCC) structure, yield strength of 393 MPa,
Young’s modulus of 207 GPa, ultimate tensile strength
of 731 MPa and the average grain size of about 90 μm.
The crack length was measured by crack-opening-
displacement gauge using the compliance method. Dur-
ing the constant-amplitude fatigue crack growth [i.e.
P
max
= 8880 N, P
min
= 89 N, a load ratio, R (P
min
/
P
max
) = 0.01, and frequency = 10 Hz], one of the
following loading conditions was applied at K =
35.90 MPa·m
1/2
. Case 1: continuously fatigue under the
same baseline condition; Case 2: a single tensile overload
(13 320 N, 150% of P
max
); Case 3: a single compressive
underload (−13 320 N); Case 4: overload–underload; and
Case 5: underload–overload. After various loading condi-
tions were applied, the constant-amplitude fatigue crack
growth tests were resumed for all cases.
A n eutron-diffraction residual stress mapping was per-
formed on L3 spectrometer at Canadian Neutron Beam
Centre, National Research Council Canada, Chalk River
Laboratories, Canada. The five compact–tension (CT)
specimens processed by the different loading conditions
[i.e. constant-amplitude fatigued (Case 1), tensile over-
loaded (Case 2), compressive underloaded (Case 3), tensile
overloaded–compressive underloaded (Case 4)andcom-
pressive underloaded–tensile overloaded (Case 5)] were
prepared to study the influence of residual stresses on the
crack growth rate, as shown in Fig. 2. Three principal
residual-strain components [i.e. longitudinal (ε
x
), trans-
verse (ε
y
) and normal (ε
z
) strains, Fig. 1a] were measured
as a function of the distance from the crack tip along the
crack-growth direction (x-direction, Fig. 1b). A total of 26
points were measured as a function of the distance from
the crack tip. To provide the required spatial resolution,
the scanning intervals of 1 mm from −4 to 0 mm (crack
tip), 0.5 mm from 0 to 8 mm where sharp strain gradients
are expected, 2 mm from 8 to 16 mm, and 3 mm from 16
to 22 mm were employed.
A schematic view of the diffraction geometry is shown
in Fig. 1c–e. For the longitudinal (ε
x
) and transverse (ε
y
)
strain components (Fig. 1c and d, respectively), the wave-
length of 1.31
˚
A was selected from the Ge115 monochro-
mator. The specimen was aligned 53
◦
(clockwise) from
the incident neutron beam and the (311) diffraction pat-
tern was measured in a stationary detector centred on a
diffraction angle of 2θ = 74
◦
. The longitudinal (ε
x
)strain
component was measured using 1-mm-wide and 2-mm-
tall (parallel to y) incident beam slits, and 1-mm-wide
diffracted beam slit. The transverse (ε
y
) strain component
was measured using 2-mm-wide and 1-mm-tall (parallel
to x) incident b eam slits, and 2-mm-wide diffracted beam
slit.
For the normal (ε
z
) strain component (Fig. 1e), the wave-
length of 1.74
˚
A was chosen from the Ge115 monochro-
mator. The specimen was aligned 127
◦
(clockwise) from
the incident neutron beam and the (311) diffraction pat-
tern was recorded in a stationary detector centred on a
diffraction angle of 2θ = 106
◦
. Thus, the diffraction vec-
tors were parallel to normal direction (parallel to z)ofthe
specimen. The incident beam was defined by 2-mm-wide
and 1-mm-tall (parallel to x) slits, and the diffracted beams
were collimated by 2-mm-wide slit.
The interplanar spacings (d-spacings) along the longitu-
dinal, transverse and normal directions were determined
from the Gaussian fitting of the (311) diffraction peak and
the lattice strains were obtained from
ε = (d − d
0
)/d
0
, (1)
c
2010 Blackwell Publishing Ltd.
Fatigue Fract Engng Mater Struct
33, 822–831
824 S. Y. LEE
et al.
Fig. 1 (a) The geometry of a Hastelloy C-2000 compact–tension specimen; (b) spatially resolved neutron-diffraction measurement positions
along the direction of crack propagation (x); schematic of diffraction geometry for the residual-stress mapping showing the scattering vector
(Q) parallel to the coordinate (c) x: longitudinal strain (ε
x
) component; (d) y: transverse strain (ε
y
) component; (e) z: normal strain (ε
z
)
component.
where d
0
is the stress-free reference d-spacing, which was
measured away from the crack tip. Three residual stress
components, σ
i
(i = x, y and z, corresponding to longi-
tudinal, transverse and normal directions, respectively),
are calculated from the three strain components using the
following equation:
σ
i
=
E
1 + ν
ε
i
+
ν
1 − 2ν
ε
x
+ ε
y
+ ε
z
, (2)
where E (= 207 GPa) is the Young’s modulus and ν (=
0.3) is the Poisson’s ratio.
RESULTS AND DISCUSSION
Figure 3 shows the experimentally measured crack-
propagation rate (da/dN) versus stress-intensity-factor
range (K) for five different loading cases. These re-
sults were previously reported,
20
and they were used to
help understand the relationship between residual-stress
distribution and crack-growth behaviour. The previous
observations are summarized as follows: Case 1 showed
a linear increase of the crack-growth rate with increas-
ing K. After Case 2 (a single tensile overload) was
introduced, the crack-growth rate was instantaneously ac-
celerated, and then a large crack-growth retardation pe-
riod was observed. Case 4 (overload–underload sequence)
showed the significantly reduced crack-growth retarda-
tion, as compared to that of Case 2. On the other hand,
after Case 3 (a single compressive underload) was intro-
duced, the crack-growth rate was initially accelerated, but
the subsequent crack-growth rate was similar to that of
Case 1.WhenCase 5 (underload–overload sequence) was
imposed, the crack-growth rates were similar to those of
Case 2, indicating a large retardation period.
To obtain a better understanding of the transient crack-
growth behaviour following the overload and/or under-
load, the residual stress fields near a fatigue-crack tip were
c
2010 Blackwell Publishing Ltd.
Fatigue Fract Engng Mater Struct
33, 822–831
NEUTRON DIFFRACTION MEASUREMENTS OF RESIDUAL STRESSES 825
Fig. 2 Neutron residual-stress mappings shown in Fig. 1 were performed on the five compact–tension specimens subjected to various
variable-amplitude fatigue-loading conditions (i.e. Case 1: constant–amplitude fatigued, Case 2: tensile overloaded, Case 3: compressive
underloaded, Case 4: tensile overloaded–compressive underloaded and Case 5: compressive underloaded–tensile overloaded). Note that
red-marked circles indicate the neutron measurement points.
measured using neutron diffraction, immediately after ap-
plying five different loading conditions, as shown in the
marked points (Fig. 2). Figure 4 shows the longitudi-
nal (σ
x
), transverse (σ
y
) and normal (σ
z
) residual-stress
profiles in the vicinity of the crack tip. In the case of
Case 1 (constant-amplitude fatigued), the tensile longitu-
dinal residual stresses were examined behind the crack tip
and the stresses were varied from tensile to compressive at
about 0.5 mm ahead of the crack tip (Fig. 4a). The normal
residual stress fields also showed similar stress distribu-
tions around the crack tip, as exhibited in Fig. 4e. The
relatively large tensile residual stresses with a maximum
of about 125 MPa were observed in a fatigue-wake region,
and the sharp transition from tensile to compressive resid-
ual stresses was examined about 1 mm ahead of the crack
tip. On the other hand, the transverse residual stresses
showed the opposite trend. The compressive residual-
stress fields with the maximum of about −70 MPa were
observed behind of the crack tip and the tensile residual
stresses were examined from about 1 to 8 mm in front of
the crack tip. The monotonic plastic zone size of about
2.5 mm was estimated from the transverse residual-stress
distribution ahead of the crack tip, as previously studied
by Rice.
23
Compared to the thickness (6.35 mm) of the
specimen, the cracks are in the predominant plane strain
condition.
After Case 2 (a single tensile overload) was applied, the
residual stress fields near the crack tip were shown in
Fig. 4a, c and e. It is noted that the application of tensile
overload yielded large compressive residual stresses near
the crack tip for the longitudinal component (Fig. 4a). For
example, the tensile longitudinal residual stresses behind
the crack tip observed in Case 1 changed into the com-
pressive residual stresses at −2.5to0mm,andthelarger
compressive residual stresses were developed at 0 (crack
tip, −123 MPa) to 3 mm. The effect of tensile overload
on the transverse residual stresses was more significant.
The large compressive residual stresses with a maximum
of −225 MPa (at 0.5 mm) were observed within ±4mm
from the crack tip. The plastic zone size of about 5 mm at
the overload was estimated from the residual stress pro-
file in front of the crack tip (Fig. 4c). A tensile overload
c
2010 Blackwell Publishing Ltd.
Fatigue Fract Engng Mater Struct
33, 822–831