Toward Predictive Understanding of Fatigue Crack Nucleation in Ni-Based Superalloys
Toward Predictive Understanding of Fatigue Crack Nucleation in Ni-Based Superalloys
FIONN P.E. DUNNE
T. BEN BRITTON
Predicting when and where materials fail is a holy grail for structural materials engineering. Development of a predictive capability in this domain will optimize the employment of existing materials, as well as rapidly enhance the uptake of new materials, especially in high-risk, high-value applications, such as aeroengines. In this article, we review and outline recent efforts within our research groups that focus on utilizing full-field measurement and calculation of micromechanical deformation in Ni-based superalloys. In paticular, we employ high spatial resolution digital image correlation (HR-DIC) to measure surface strains and a high-angular resolution electron backscatter diffraction technique (HR-EBSD) to measure elastic distortion, and we combine these with crystal plasticity finite element (CPFE) modeling. We target our studies within a system of samples that includes single, oligo, and polycrystals where the boundary conditions, microstructure, and loading configuration are precisely controlled. Coupling of experiment and simulation in this manner enables enhanced understanding of crystal plasticity, as demonstrated with case studies in deformation compatibility; spatial distributions of slip evolution; deformation patterning around microstructural defects; and ultimately development of predictive capability that probes the location of microstructurally sensitive fatigue cracks. We believe that these studies present a careful calibration and validation of our experimental and simulation-based approaches and pave the way toward new understanding of crack formation in engineering alloys.
Materials science and engineering focus on
understanding how defects and microstructure control
performance. In structural materials, our focus is on
strength and failure, and the role of grains is
critical. Deformation processes at the grain level
are captured within the domain of crystal plasticity,
and these processes can be characterized through
experiment and simulation. In engineering alloys,
the evolution of the microstructure, such as grains,
damage, or dislocation structures, is critical in
many significant industrial problems, such as
fatigue crack nucleation.1–3
Recent progress in understanding the evolution of
the microstructure during mechanical deformation
has been afforded through the exploitation of direct
transmission electron microscope (TEM) dislocation
characterization. Experimental results from TEM
are qualitatively beautiful and often quantitatively
useful, such as the nature and structure of
dislocations within a dislocation wall4,5 or the dislocation
types, spacings, and reactions involved in a
dislocation grain–boundary interaction.6,7 These
observations are useful snapshots of local behavior,
but the volume of material probed is necessarily
limited,* and therefore, capturing the grain
behavior at longer length scales and from bulk samples is
of limited use.
*Assuming that it is possible to study a 100-nm-thick 10 9
10lm2 foils in one half day of TEM time, it would take a billion years
of TEM time to study the volume of material contained within a
The performance of engineering alloys often
involves understanding deformation across multiple
length scales and often the interaction of
deformation near heterogeneous interfaces such as grain
boundaries. This level of detail is essential to predict
component performance, and therefore, features at
the micrometer length scale are often critical, such
as for microstructurally sensitive processes involved
in crack nucleation. This motivates extension of
insight at the nm length scale, obtained from TEM
thin foils extracted from lightly deformed samples,
to studies that include understanding of
deformation across multiple grains, to higher strains and at
larger length scales.
In the past five years, significant progress has
been made to improve our understanding of
plasticity in polycrystals through the development of
new experimental techniques, e.g., cross-correlation
based high angular resolution (HR-EBSD)8,9 and
high spatial resolution digital image correlation
(HR-DIC),10–13 which provide the means to measure
elastic and plastic deformation gradients within
polycrystals, including elastic strain, lattice
rotation, plastic strain, and continuum rotation.
The results obtained can be used directly to
validate the developed crystal plasticity finite
element (CPFE) modeling results.14–16 The
combination of these micromechanical techniques provides a
powerful tool to reveal the underlying deformation
mechanisms within crystals and enhances our
understanding of plasticity in polycrystals. The
aim of this review article is to summarize and draw
together the micromechanical studies carried out
within our research group based on these three key
techniques to address the fatigue crack nucleation
CHARACTERIZATION, MODELING, AND
HR-EBSD enables the determination of elastic
strain and the lattice rotation tensors based on
polar decomposition.17 It also enables the study
of geometrically necessary dislocation (GND)
structures, that is, those dislocations that give
rise to a net Burger closure failure. Lattice
rotation gradient fields are employed to estimate
the GND content stored within crystals,18,19
providing improved angular resolution ( 1009)
over the conventional Hough-based method.19
The HR-EBSD can also be used to estimate total
dislocation density through statistical analysis of
distribution of shear stress within a probed
Digital image correlation (DIC) is an optical
method to spatially resolve displacements of
features across the surface of a sample. The image
analysis technique is used to track these
displacements with high precision, through analysis of
regions of interest (ROIs), ultimately giving strain
maps of a high resolution. We have developed a
simple and effective surface coating method that
was first reported by Jiang et al.17 where nanosize
( 250 nm) silica particles were stably and
uniformly applied onto the free surface of the
samples.17,21,22 These nanosize particles enable spatial
high-resolution strain measurements as these
highcontrast features are imaged at high-image
magnifications. SEM-based imaging of the surface
displacement field using our in-house MATLAB-based
DIC codes enables us to obtain high-resolution,
fullfield strain maps.
Crystal plasticity finite element methods model
true representations of the single-, oligo-, and
polycrystal Ni samples. The models have been
well reported on in the literature and, hence, are
not elaborated on here, but readers may find
details in Refs. 14, 15, and 23. The crystal slip rule
is as follows:
where the shear strain rate on each slip system, c_i,
is linked through a sinh term to slip activity when
the shear stress on the slip system, si, exceeds the
critical resolved shear stress, sc, subject to an
activation volume DVi and thermal contributions
with Boltzmann’s constant multiplied by
temperature, i.e., kT. The thermal activity of crystal slip is
encompassed within an exponential, describing a
Gibbs-based statistical approach in capturing the
average glide velocity, where DF is an activation
energy for the obstacle type. Finally, the prefactors
qgbi2 m capture the density of glissile dislocations qg,
their Burgers vector bi, and the dislocation jump
frequency m. Many of the required properties are
now directly measurable through the use of
Mechanical testing was performed on
single-crystal CMSX4, oligocrystal MAR-002, and powder
metallurgy FGH96 and RR1000 Ni-based
superalloys under three-point bend testing, which focuses
the maximum tensile stress at a known location for
HR-DIC and HR-EBSD analyses.
Single- and Oligocrystal Behavior
Our first case study addresses the slip in
singlecrystal samples, which was subjected to an
incrementally increasing three-point bending load (as
seen in Fig. 1a). The slip activities on the free
surface were captured by an optical microscope
(OM) as revealed in Figs. 1 and 2, and a
microstructurally faithful CPFE model was created to model
the plasticity for the given crystal orientation.
From the experiment, two major slip fields are
observed: (I) one located within the major tensile
region and the extent of this field varies along the
length of the beam, as the sample is deformed under
Fig. 1. (a) Schematic diagrams showing one single-crystal, three-point beam loading the key regions of interest for slip, the predominant slip
activity, the locations and beam faces at which HR-DIC and HR-EBSD characterization have been carried out, and finally, the progressively
increasing load cycles applied. (b) Identification of FCC slip directions and normal, and slip direction number designations used throughout this
section of the article, shown with respect to an orthogonal RHS (x, y, z) reference system (figure adapted from Ref. 25).
Fig. 2. (a) Comparison of the experimentally (HR-DIC) measured and the crystal plasticity predicted slip fields for sample 1 on the DIC face. Slip
systems 3 and 10 shown in Fig. 1 are found to predominate, and their spatial distributions are well captured by the crystal plasticity model. (b)
The HR-DIC measured and the CPFE determined accumulated plastic strain corresponding to the cycles shown, obtained by averaging the
measured and the predicted strains over the region of interest (figure adapted from Ref. 25).
three-point bending; and (II) horizontal slip bands
operating near the apparent neutral axis on the
right of the imaged beam section.
The shape and extent of these fields were well
captured by the CPFE modeling, and the results are
compared with the OM observed slip activities in
Fig. 2a. From the CPFE model, it is apparent that
the anisotropic nature of crystal deformation is
essential in capturing the nature of plastic slip,
even in this simple test, demonstrated by the second
slip system activation. The second validation of the
operation of these slip systems has been confirmed
by using slip trace analysis, and the active slip
systems matched well with the CPFE predictions.
The HR-DIC was used to measure the strain field
in a region of interest (ROI) of 250 lm 9 250 lm
size at the middle bottom region of the sample. As
revealed in Fig. 2a, captured slip lines roughened
and broadened with increasing load. The explicit
heterogeneity of dislocation slip (i.e., discrete slip
bands) at this length scale is necessarily smoothed
out within the CPFE approach, and capturing this
would need discrete dislocation modeling.
Nevertheless, CPFE can capture the slip fields and the
predicted effective strains that agree with those
averaged from the HR-DIC measurements (see
Fig. 2b). It should be noted that the elliptical ring
surface feature in the OM image in Fig. 2a is an
artefact as a result of sample preparation. An
enlarged OM map of the sample can be found in
Ref. 25. The fact that we capture the size, extent,
and slip system type for the slip fields in both
simulation and experiment, using a single-crystal
exemplar, gives confidence about the accuracy of
A more complex challenge is to capture the
heterogeneous nature of a crystal slip in a sample
with multiple grains. An oligocrystal sample,
consisting of six grains within the ROI, is used. The
CPFE model is shown in Fig. 3a. Individual
inFig. 3. (a) A large-grained Ni oligocrystal sample with a region of interest (ROI) identified and crystallographic orientations of grains 1, 2, and 3
shown, subject to three-point cyclic bend loading shown and the finite element discretization employed for the ROI. The insets show direct
comparisons of the HR-DIC and the CPFE in-plane rotation measurement. The associated quantitative comparisons of experiment and
simulation of the marked lines CC¢ and DD¢ are revealed in (b) (figure adapted from Ref. 25).
plane plastic strain and rotation terms measured by
HR-DIC at the ROI were compared with the CPFE
predicted results. Quantitative and qualitative
agreement between experiment and simulation is
excellent, as shown by the HR-DIC experiments and
CPFE simulations (Fig. 3b).
HR-EBSD Study of Plasticity in Polycrystals
Containing a Second Phase
Nonmetallic inclusions are inevitably mixed into
the Ni-based superalloy powders during the powder
metallurgy (PM) forming process used to make
components, such as turbine disks,26,27 and are
often found to be the fatigue crack initiation
sites28,29 and limit fatigue life of PM formed
components. Inclusion sizes are carefully controlled by
filtering out the large inclusions30 so that the
remaining inclusion sizes are comparable with
nickel powder particles, which are approximately
the size of several grains.
We have conducted two case studies, one which is
focused on following the evolution of the strain state
using HR-EBSD, and the second is focused on
exploring the surface strain state with HR-DIC.
We have cut samples to locate an inclusion within
the characterized ROI. Three-point bending tests
were carried out with increasing load (HR-DIC
sample) and an increasing number of fatigue cycles
The results from the HR-EBSD study are
presented in Fig. 4, which shows characteristic SEM
micrographs of this area and that changes in
surface topography are microstructurally sensitive
as a result of the evolution of the surface slip, shear
along the twin boundaries (revealing a stepped
morphology with a frequency related to the twin
structure), and all these features are observed after
only two cycles. A small crack occurred at the
inclusion after 20 cycles, which continued to grow
through the matrix. Longer cracks (>8 grains) were
formed after 5200 cycles.
The evolution of the map-averaged GND and total
dislocation density are shown in Fig. 4c. From the
total GND density plot, the most striking
observation is that the GND density rapidly increases and
then shows an apparent decrease. This decrease
could be a result of plastic shakedown, where
initially high dislocation densities are reduced
because of the formation of lower energy structures.
Nevertheless, further investigations31 on this
sample revealed that there was a systematic reduction
in GND density in the locally imaged area, as a
result of the growth of a carbon film during imaging,
which reduces the sensitivity of the GND
measurement, and so the reduction of GND density with
cycles should be investigated with care. In addition,
the HR-EBSD approach to estimating total
dislocation density is based on linking the distribution of
stresses probed to a distribution of edge
dislocations, while the screw type of dislocations is
neglected. To address this problem, ECCI is a
promising technique to be used as an alternative
dislocation characterization method to determine
the total dislocation density.32,33
The initial dislocation density was low and
homogeneously distributed through the microstructure,
and the presence of residual stress gradients was
low.31 After two cycles, the GND density and
residual stress gradient ‘‘hot spot’’ (red) maps are
shown in Fig. 5, where points from the top 5% of
GND density distribution from the entire map are
overlaid on the EBSD image quality (IQ) map.
‘‘String’’ patterns of these hot spots are found to
correlate strongly with the formation of the
underlying cracks. From this study, it is unclear whether
the presence of a high GND density is the precursor
Fig. 4. (a) An EBSD crystal orientation map of the ROI with respect to the major deformation axis. (b) A SEM image showing the inclusion and
fatigue cracks near it. (c) A statistical analysis of GND and total dislocation density development as a function of the number of fatigue cycles
(figure adapted from Ref. 31).
Fig. 5. GND density map (a) and grain normalized in-plane shear stress map (b) after two cycles. The dark gray lines represent grain
boundaries. Points from the top 5% of GND density distribution, from the entire map, after cycle two are plotted as red dots overlaid on the EBSD
image quality map in which the crack path is clearly marked (figure adapted from Ref. 31).
to crack formation or whether the high GND density
is forming where the grains are being ‘‘pulled apart’’
the most by their local neighborhood.
HR-DIC and HR-EBSD Study of the Plasticity
in Polycrystals with Inclusions
The second sample contains an inclusion in the
ROI as shown in Fig. 6. The HR-EBSD analysis was
carried out at the beginning and end of the loading
process. After each loading cycle, the sample was
removed from the rig and placed in a SEM to
acquire high-quality micrographs for the HR-DIC
analysis. Thus, in this test, we obtained both plastic
and elastic distortions at the same region where
cracks were formed. Combining with the in situ
HREBSD analysis mentioned earlier, we were able to
link the evolution of elastic and plastic distortions in
polycrystals to examine the crack formation process.
The effective strain distribution and evolution are
shown in Fig. 6c. Similar to the GND density, the
dominant structures within the effective strain
maps were formed at the early stage of plasticity.
As expected, the map-averaged effective strain
values increased with increasing load (this is
different to the GND density evolution from the
previous example) and the extent of heterogeneity
gradually increased as indicated by the error bars in
Fig. 6c and d. The effective strain distribution is
strongly localized around the inclusion, and it is
formed into a ‘‘butterfly’’ shape that is sensitive to
both inclusion and the microstructure.
Six micro-cracks within the matrix were observed
at locations shown in Fig. 7a. The experimentally
measured accumulated plastic slip, GND density,
and isotropic elasticity finite element simulation of
maximum shear stress at the same area are shown
in Fig. 7b, c, and d, respectively. In comparing these
three maps, it is interesting to see that the crack
sites have high accumulated slip, high maximum
shear stress, and high GND density. It is likely that
the evolution of these fields and the cracking are
related. Yet, detailed inspection of each crack with
respect to each field reveals that they are not
independent indicators of crack initiation as, for
instance, the GND density maps show hot spots
toward the interior and away from the inclusion
where cracking was not observed. The effective
strain map seems to be a more reasonable
Fig. 6. (a) An initial EBSD inverse pole figure map near inclusion with respect to deformation axis. (b) The HR-DIC measured strain distribution
near the inclusion at 5900N. (c) The statistical analysis of effective strain distribution and development as a function of load. Three examples of
detailed strain distribution are plotted as a histogram and shown in (d) (figure adapted from Ref. 21).
Fig. 7. (a) A SEM image shows that six micro-cracks were formed in the nickel matrix at various locations after applied 5900N as highlighted with
red circles. (b) The effective strain map overlaid with grain boundary map determined by HR-DIC and EBSD, respectively. (c) The lower bound
estimated GND density maps at cracked states (5900N). (d) Finite element analysis of the max shear stress distribution at the inclusion. The
nonmetallic inclusion (E = 50 GPa, v = 0.31) was considered to be softer than the nickel matrix (E = 200 GPa, v = 0.31) (figure adapted from
independent map, and this is supported by a
previous study by Dunne et al.34 in which they
found that the accumulated slip can accurately
predict the fatigue crack nucleation sites for most of
the studied samples. Nevertheless, in this previous
study, not all cracks could be accommodated
through a simple evaluation of the accumulated
plastic strain (i.e., slip) alone, and so a better match
was achieved with a stored energy density criterion
for fatigue crack nucleation.
Use of the new criterion allows for consideration
of the local stored energy rate (per loading cycle)
developed over an area determined by the local
dislocation content to define the appropriate length
Fig. 8. (a) The agglomerate inclusion model and the corresponding paths 1 along which xx strains are obtained by the HR-DIC and from the
CPFE simulation extracted for (b) path 1 at the end of the second cycle shown figure inset in (a). The colors in (a) are used to distinguish grains.
(c) The CPFE predicted maximum principal stress distribution. (d) A plot of CPFE determined normal stress as a function of effective plastic strain
for the observed inclusions (figure adapted from Ref. 35).
scale with which to define the energy density. It,
therefore, places emphasis on the importance of
(Griffith-like) stored energy and dislocation density
(Stroh’s model). Dunne et al.34 posited their model
through the following argument:
They considered storage volume DV, written in
terms of storage area DA, statistically stored
dislocation (SSD) and GND density, which is given by:
The stored energy per cycle within the volume is
then determined to be where n is the fraction of the
dissipated energy stored in the establishment of
dislocation structure, and the integration is carried
out over a complete loading cycle. The stored energy
rate per cycle is then given by:
For this study, Eq. 4 must be simplified to enable
an indication of the stored energy rate to be
estimated from our experimental observations. We
do not have a measure of the local absolute stress
variation, and so the stress is assumed to be
constant and uniform according to the tensile
boundary conditions because the local
microstructural stress around a given cyclic hysteresis loop is
not known. Note that Fig. 7d is a finite element
model prediction that does not have explicit local
microstructural representation so that the stresses
calculated in this work are from assuming Mises
plasticity. Although this may not be a good
representation at localized regions, e.g., grain
boundaries, it is likely to be reasonable for the
grainaverage level and is therefore a tolerable
Also, we consider that the SSD density qSSD is
proportional to the applied plastic strain e.14 We
therefore only use the GND density as a stronger
influence on the distribution of stored energy
This reduces Eq. 4 to give:
I I : dep
The variation of this parameter was found to have
a better correlation with the crack locations as
compared with either accumulated plastic strain or
accumulated GND density independently.
Explicit CPFE modeling of the inclusion and
surrounding microstructure is underway, which
should provide more insights on the stress
distribution near crack nucleation sites and should allow
a better examination of this crack nucleation
CPFE Model of Polycrystal Containing
Second-Phase Particle Fracture and
The third sample that contained a distribution of
nonmetallic inclusions within the Ni matrix was
tested and studied by HR-DIC and CPFE.33 This
inclusion was generated as a result of a different
processing route, and so the microstructure and
properties of this inclusion structure are different
than those discussed, as observed in the explicit
microstructure rendered for the CPFE study shown
in Fig. 8a.
The comparison of the surface strain fields
between each grain surrounding the inclusion
captured by HR-DIC and simulated by CPFE, shown in
Fig. 8, reveals good agreement between model and
experiment and enables the study of the inclusion
effects on local plasticity and fatigue crack
The experimental observations revealed both
decohesion of the matrix and fracture of the
particles. As the CPFE model matched the total strain
fields from the experiment, the stress fields
predicted with the CPFE were used to understand the
nature of decohesion and failure of the particles and
nearby matrix. The full-field maps representing
different reductions of the stress tensor for each
point within the modeled region were created, and
points near the particles that failed in the
experiment were highlighted. The evaluation of the
hydrostatic and normal stress fields (and many
others that were not reported within the article)
revealed that the normal stress perpendicular to the
inclusion-matrix interface was found to show the
strongest ‘‘contrast’’ and best correlation.
The evaluation of the stress differences between
the particles that had failed and those that had not
enabled us to predict that the interfacial failure
stress for these particles was 1270 MPa (as
observed in Ref. 35 in Fig. 14c), which is a result
possible only through combined CPFE and HR-DIC
SUMMARY AND FUTURE PERSPECTIVES
The aim of this research effort was to develop
predictive capability for fatigue crack nucleation in
polycrystalline metallic materials. We designed our
series of experiments to build complexity step by
step and to validate each microstructural feature
and experimental technique in turn and have
ultimately achieved a good match between
experiment and simulation for crack nucleation near
inclusions in polycrystal Ni-based superalloys. The
correlative approach that comprises a controlled
series of increasingly complex microstructures
coupled with HR-EBSD, HR-DIC, and crystal plasticity
simulations has enabled calibration and validation
of each approach.
A significant body of evidence from direct
measurement has now been established for Ni single,
oligo-, and polycrystals, as well as for polycrystals
containing agglomerates, that no single
microstructural quantity, be it local slip accumulation, type III
stress, or density of GNDs, in its own right provides a
persuasive link to fatigue crack nucleation. Although
the development of local slip has been found to be a
prerequisite for fatigue crack nucleation, in keeping
with much earlier work for the need for persistent
slip band (PSB) formation, it is argued to be a
necessary rather than a sufficient process. We have
found many instances, however, within the
experimental programs reported, where highly localized
slip does not give rise to crack nucleation.
Nonetheless, the contemporaneous utilization of HR-EBSD,
HR-DIC, and CPFE modeling has made clear that the
highly anisotropic activation of slip, its
inhomogeneous development, and its distribution can be
effectively captured with CP modeling. Hence, highly
localized slip accumulations, progressively
increasing over cyclic loading, local type III stresses, and
GND densities (within the constraints outlined
earlier) are beginning to become quantitatively
predictable such that refocus on quantitative prediction
of fatigue crack nucleation is appropriate. A new
quantity that seems to have merit is found to be a
local stored energy density discussed in this article.
The evidence presented is limited to the Ni
agglomerate polycrystal where the multiple locations of
crack nucleation can be captured with the stored
energy approach. Nevertheless, the results of other
continuing work have revealed that the locations of
crack nucleation can be correctly captured in single
and oligocrystals and that the experimentally
observed scatter in cycles to fatigue crack nucleation
can also be captured with this approach in
largegrained Ni RS5 alloy.36
Equally important as fatigue crack nucleation, but
as of yet elusive, is the establishment of quantitative
mechanistic understanding of the processes by
which microstructurally sensitive fatigue cracks,
once nucleated, subsequently grow. The paths they
take, be they in transgranular or intergranular (and
often both) modes, the rates at which growth occurs
in these modes, taking due account of the highly
anisotropic nature of transgranular growth in, for
example, HCP crystals, all remain to be understood.
The role of interfaces, grain or twin boundaries, or
phase interfaces all influence crack growth rates and
remain to be explained. These are some of the many
challenges that remain and provide interesting
opportunities for future research.
As shown in this study, the understanding of
plasticity in polycrystals at room temperature has
been significantly improved through the HR-EBSD,
HR-DIC, and CPFE techniques. In-service
operating conditions of high-value materials such as
aeroengine materials are often at elevated
temperatures, which involve viscoplasticity where creep will
also be an important deformation mode to be
considered. Therefore, the characterization and
modeling tools need to be further developed to
measure and predict crystal behaviors at high
We thank Dr. Mitch Cuddihy for assistance in
typesetting of the CP-FEA formulation. We thank
Iain Parr and Mark Hardy for useful discussions.
TBB and FPED would like to acknowledge the
Royal Academy of Engineering for funding toward
their respective Research Fellowship and Research
Chair. We acknowledge support for our ‘‘Nickel
Campaign’’ from multiple sources over its
continuing duration, including Rolls-Royce PLC, the
Beijing Institute of Aeronautical Materials, and EPSRC
(EP/K034332/1 and EP/L025213/1). A significant
fraction of the underpinning work reported within
this article was conducted within the AVIC Centre
for Materials Characterisation.
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