# Microstructure sensitive fatigue life prediction model for SLM fabricated
Hastelloy-X

Chandrashekhar M. Pilgar  Ana Fernandez  Javier Segurado

###### Abstract

A microstructure-sensitive fatigue life prediction framework based on CP-FFT
is proposed to study SLM fabricated Hastelloy-X. The microstructure enters in
the model through the shape, size and orientation distributions of grains in
the RVEs, which are generated from experimental EBSD data. The framework has
been applied to specimens built in two different directions with different
polycrystalline microstructures and is able to accurately predict their
fatigue life, using just two experiments for calibration. The model reproduces
the better fatigue performance found experimentally at high stresses for
samples built in transverse direction, identifying the origin of this
anisotropy in grain aspect ratios.

###### keywords:

††journal: International Journal of Fatigue

[inst1]organization=IMDEA Materials Institute,city= C/Eric Kandel 2,
postcode=28906, state=Getafe,Madrid, country=Spain

[inst2]organization=Universidad Politécnica de Madrid, Department of Materials
Science,addressline=E.T.S.I. Caminos, postcode=28040, state=Madrid,
country=Spain

[inst3]organization=Industria de Turbo Propulsores, Department of Materials
and Processes, ITP Aero SAU,postcode=48170, city=Zamudio, state=Bizkaia,
country=Spain

##  1 Introduction

Hastelloy-X is a nickel-based superalloy, primarily strengthened by adding
solute elements W, Cr, and Mo [1, 2] which shows excellent oxidation and
corrosion resistance at elevated temperatures, making it suitable for gas
turbines and jet engine parts. Moreover, the alloy shows good formability and
weldability, making it an ideal candidate for additive manufacturing
fabrication, mainly using a selective laser melting (SLM) route [2].

SLM is an additive manufacturing process that can achieve near net shapes of
complex geometries with stable properties. In this process, a laser beam
follows a predefined 2D path over a platform in a sequential manner. After one
traverse of a laser beam, the platform is lowered by the predefined layer
thickness. This procedure is repeated till the completion of the final 3D
object [3]. SLM fabrication allows tailoring of the microstructure by changing
parameters such as laser energy [3], scanning strategy [2, 4] or build/sample
geometries [2, 4]. The production of net-shape complex components of
Hastelloy-X using a SLM process presents enormous potential for the industry,
especially aerospace, which have already started to design and fabricate some
jet engine parts using SLM fabricated Hastelloy-X. However, strong variability
is found in the monotonic and cyclic/fatigue behavior of SLM components
depending on the process parameters and part geometry. This variability of
properties is mainly attributed to the highly heterogeneous microstructures
resulting from the complex thermal cycles followed during fabrication [5, 6,
7, 8] . The most critical design parameter for potential high-temperature
applications of SLM fabricated Hastelloy-X parts is the fatigue performance.
Therefore, models able to predict the fatigue life for SLM components
accounting for the effect the heterogeneous microstructure are fundamental for
the progressive introduction of these components into industrial applications.

Multiple authors have studied the cyclic deformation and fatigue behavior of
SLM components from an experimental viewpoint [9, 10, 11, 5, 12, 13, 14]. Most
of these studies reported similar conclusions for a wide range of SLM alloys
including IN718 or Ti-6Al-4V [5, 15], namely an inferior fatigue performance
of the SLM component compared to wrought and cast ones. In some cases, the
reduction of fatigue life can be larger than one order of magnitude [14, 16,
15, 7]. This worse fatigue performance of SLM components, particularly in as-
built case, has been attributed to the porosity [17], surface roughness [18]
or residual stresses [19], and also to the resulting anisotropy of the part
and its orientation with respect to the cyclic loading direction [19, 15].
However, some studies reported the mitigation methodologies for porosity,
surface roughness and residual stresses effects on the fatigue lives by
optimizing the processing parameters[20], heat treatment[21] and powder
composition[3]. With respect to the specimen orientation, the fatigue life of
vertically built specimens has been found to be inferior to horizontally built
parts when loaded under stress control [22, 16, 12, 23, 19]. On the contrary,
under strain-controlled loading, some studies [10] show the opposite effect,
vertically built samples having a superior fatigue performance than
horizontally built samples. This fact is justified by the resulting elastic
anisotropy, vertically built samples have lower elastic modulus and hence
underwent lower stress amplitudes than horizontally built samples.

In the particular case of AM Hastelloy-X, a few recent studies can be found in
the literature which cover its fatigue analysis. Han et al. [24] studied the
effect of hot isostatic pressing (HIP) on the fatigue life of AM Hastelloy-X
and concluded that the fatigue life could be significantly improved by HIP due
to closure of internal pores and relief of residual stresses. Esmaeilizadeh et
al.[25] studied the fatigue life of AM Hastelloy-X as a function of laser
scanning speed at room temperature and concluded that fatigue lives could be
improved by optimizing the processing parameters. Wang et al.[26] highlighted
the importance of HIP on fatigue life improvement and reported that the part
orientation does not play a significant role in the fatigue life for low
applied stress level; however, at higher loads, observed that the fatigue
resistance of vertically orientated specimen is inferior to its horizontally
oriented counterpart. Lindstrom et al.[23] reported that the fracture surface
of the horizontally oriented specimen had shown more plastic deformation than
the vertically oriented specimen.

From the modelling view point, empirical models such as Basquin, Coffin
Manson, etc., have been used with experimental data to correlate the fatigue
life of different additive manufactured alloys [27, 28, 29], including the
study of SLM Hastelloy-X by Esmaelizadeh et al.[20]. However, it is well known
that there is a strong influence of the microstructure in the fatigue life of
an alloy, and these models cannot account for it because they are empirical
and calibrated with experimental data without considering any microstructural
aspect.

The influence of the microstructure in the fatigue performance is mainly
affecting the nucleation of a crack. The repeated deformation of the
polycrystal induces the localization of plasticity at the grain level in
persistent slip bands which degenerate finally in microscopic fatigue cracks
[30]. The regions of the microstructure in which these bands and subsequent
cracks are more prone to appear correlate with the points in which some
variable related with cyclic plastic deformation is localizing. In order to
consider the effect of the microstructures resulting from SLM fabrication in
the fatigue life prediction, micromechanical models which model this
mechanisms have to be used.

In particular, the approaches relying on the use of polycrystalline
computational homogenization [31] to estimate the number of cycles for crack
nucleation [30] allow to analyze crack nucleation as a function of the
microstructure from a statistical viewpoint. Under the microstructure
sensitive fatigue life approach, fatigue life is predicted by cyclic
accumulation of plastic slip and localized stresses that act as a driving
force for fatigue crack nucleation on persistent slip bands[32]. The driving
force is quantified using Fatigue Indicator Parameters (FIP) that are based on
accumulated plastic slip [33, 34], stored strain energy [35, 36, 37],
dissipated energy [38, 39, 40, 41] or in general any combination of stress,
strain and plastic slip fields [42].

The FIP is evaluated at each material point by varying local micro-fields and
internal variables, and fatigue life is calculated based on the maximum FIP
(hot-spots) in the domain. These models include the relevant microstructural
features of the polycrystal under study into Representative Volume Elements
(RVE) of the microstructure. The mechanical response of the RVEs under a
macroscopic cyclic history is then simulated using crystal plasticity and some
computational homogenization approach to obtain some fatigue indicator
parameter that is the driving force for crack nucleation. These
micromechanics-based fatigue life prediction models have been successfully
applied to a large number of alloys produced by standard fabrication routes,
as wrought Ni-based superalloys [33, 43, 40]. Most of the micromechanical
studies of fatigue rely on finite element simulation and only recently Fast
Fourier Transform (FFT) solvers have been introduced as an alternative to FEM
[42, 44, 32]. The use of FFT-homogenization methods allow accurate prediction
of the micro-field response over many cycles at a fraction of the cost of FE
based solvers, making it possible to simulate larger RVEs and consider more
accurate microstructural features [44].

The application of crystal plasticity models to investigate the effect of the
resulting microstructures of AM metals in their fatigue performance is a
trending topic nowadays. These models have already been applied to some SLM
fabricated superalloys [45, 46, 47, 48]. In some cases the life prediction is
focused on crack propagation [47] stage. Other studies consider crack
nucleation but the life prediction model is directly linked with the
macroscopic cyclic response [25, 20, 27], without making use of the
distribution of microscopic fields obtained. Only a few studies on AM alloys
are available which rely on the use of micromechanical fields through fatigue
indicator parameters (FIPs) to estimate the fatigue life assuming crack
nucleation controlled fatigue. Among these studies, in some cases the real
microstructures are not considered and simplified random textures are assumed
[49, 50]. In other cases, the actual microstructures obtained from EBSD data
are included in the model [51, 46, 52, 53] but the study is focused only on a
particular fatigue regime, either LCF [46, 51, 53] or HCF [54, 55], and a
particular loading control, either strain or stress control.

To summarize, on one hand, there are no available micromechanics based fatigue
life prediction approaches available which consider both strain and stress
control and a wide range of fatigue lives. On the other hand, there is still a
lack of understanding of the effect that the building direction has on the
fatigue performance of SLM alloys, originated in the different microstructures
developed during SLM fabrication.

In this work, we present a micromechanics based fatigue life prediction
approach valid for stress and strain control and a wide range of fatigue
lives. From a numerical viewpoint, FFT based polycrystalline homogenization is
used to simulate the cyclic response of the specimen. This framework is then
applied to analyze the effect of the SLM resulting microstructures, including
grain aspect ratios and orientation distributions. The model is applied to
study the fatigue life of Hastelloy-X specimens built in different directions
[56] including its validation using experimental data under strain and stress
control at 750 ∘C for a wide range of fatigue cycles from very low LCF to more
than $10^{5}$ cycles.

##  2 Experimental study

###  2.1 Material and fabrication

This work uses a selective laser melting(SLM) route to manufacture the
Hastelloy-X samples, whose nominal composition is given in [3].

The bulk samples were built layerwise in the Z direction using the Renishaw
RenAM 500Q machine, keeping a constant layer thickness of 60 $\mu m$ and a
rotation of the laser scanning direction between consecutive layers of 67∘
(see Fig. 1). Hastelloy-X powder was made via the Ar gas atomization process,
providing a powder size in the range of $15\;\mu m$ to $53\;\mu m$. The
parameters of the SLM process, such as laser power, scan speed, hatch
distance, and volumetric energy density, were well optimized to achieve the
maximum part density. The different fatigue specimens were fabricated through
the SLM process by ITP Aero in the shape of bulk blocks. The bulk samples were
fabricated in two different orientations (see Fig.1):

  * 1.

Bulk Z specimen is built in a cylindrical shape, oriented parallel to the
build direction.

  * 2.

Bulk X specimen is printed with a rectangular shape with dimensions 15
mm$\times$15 mm$\times$100 mm, oriented perpendicular to the build direction.

Coupons necessary for fatigue tests were machined from above-built bulk
samples in cylindrical dog-bone shapes with the same gauge length (12.7 mm)
and diameter (5.08 mm) shown in Fig.2. Finally, the samples were heat treated
at 1170∘C / 30 minutes and followed by gas cooling. The SLM thermal history
was removed by this treatment and the SLM process dendrites were dissolved.
This annealing also significantly minimises the residual stresses of
Hastelloy-X, as observed in Karapuzha et al. [57]. It is also interesting to
note that precipitates were found in significant quantities in the Ni-based
as-build samples, but after the heat treatment the presence of precipitates in
the alloy was strongly reduced or even suppressed.

Figure 1: The SLM platform shows different orientations of the bulk samples
fabricated using the SLM process along with the build direction. Figure 2:
The cylindrical dog-bone-shaped specimen, used for the uniaxial tensile
fatigue test.

###  2.2 Microstructure

The resulting porosity was obtained using a microCT analysis and was below 1%
in all the samples. With respect to the polycrystalline microstructure, it is
well known that changes in the printing process parameters generate
microstructures with different grain elongation and textures. The bulk samples
showed different grain structures in BD than in the XY plane. The elongated
grains are observed in the build direction. The direction of epitaxial grain
growth depends on the direction of maximum heat flow. The maximum heat flow in
the SLM process is observed in the building direction. However, the equiaxed
grains are observed in the lateral direction.

The grain size, shape, and orientation distributions were obtained using
Electron Backscatter Diffraction(EBSD) imaging technique. Concerning the grain
morphology, the Bulk Z sample reveals elongated grain arrangement in the build
direction along with an equiaxed grain setup in the XY section (see Fig. 4).
The Bulk X specimen exhibits more equiaxed grain arrangement in both sections.
The average grain aspect ratio was 1/1/2 and 1/1/1.5 in X-Y-Z axis, for Bulk-Z
and Bulk-X samples respectively.

The diameter of the equiaxed grains in the XY plane follows the log-normal
distribution. The log-normal distribution parameters(mean and standard
deviation) for both bulk samples are enlisted in Table 1. The surface
roughness is always present in the SLM samples due to partially melted and
unmelted powder on the surface. Therefore, the samples are machined from all
directions to minimize the surface roughness impact on the fatigue test.

Table 1: The lognormal distribution of grain diameter of the bulk samples Sample | Bulk Z | Bulk X
---|---|---
Average($\mu m$) | 17.94 | 20.13
Standard deviation | 0.94 | 0.76

To obtain the orientation distribution of each grain in the microstructure,
the EBSD analysis was performed on the samples before the fatigue testing. For
this purpose, a SEM with an EBSD detector was used. A confidence index (CI)
criteria was used to ensure the accuracy of the measured Euler angles. The
Euler angles are measured only if the CI at a pixel is greater than 0.02. The
OIM analysis software generates the resultant EBSD orientation maps and M-TEX
[58] generates corresponding pole figures in Z-direction, which are depicted
in Fig.3 and 4 for the XY and XZ sections of Bulk Z and Bulk X samples
respectively. The colors in the EBSD maps that follow the IPF triangle gives
the orientation of the building direction concerning the crystal reference
frame, and the colors in the pole figures follow the color bar shown at the
bottom of Fig.3 and 4. The pole figures depict that both bulk samples have a
random texture along the XY and XZ sections.

Figure 3:  EBSD images of Bulk Z (top) specimen in XY and XZ sections
respectively. The colors of the maps suggest the orientation of the building
direction compared to the reference frames following the IPF triangle. Pole
figures using all the orientations (second) and a reduced set obtained after
sampling the ODF(bottom), for Bulk Z specimen. Figure 4:  EBSD images of Bulk
X (top) specimen in XY and XZ sections respectively. The colors of the maps
suggest the orientation of the building direction compared to the reference
frames following the IPF triangle. Pole figures using all the orientations
(second) and a reduced set obtained after sampling the ODF(bottom), for Bulk X
specimen.

###  2.3 Mechanical Characterization

The uniaxial cyclic performance of SLM fabricated Hastelloy-X at 750∘ was
experimentally determined by cyclic tests performed in a wide range of strain
and stress amplitudes. The tests were carried out according to the standards
PrEN3874-98 and ASTM E606. The results of the fatigue testing campaign will be
presented in the result section.

##  3 Micromechanics Based Fatigue Life Prediction Model

A computational polycrystalline homogenization framework based on the crystal
plasticity and FFT homogenization (CP-FFT) is used to simulate the cyclic
response for SLM Hastelloy-X under strain and stress-controlled loading. The
FFT-based homogenization code FFTMAD [59] is used to simulate the cyclic
response of Representative Volume Elements (RVE) of the microstructures
considered.

The synthetic RVEs contain grain size distribution, grain aspect ratio, and
orientations representative of the actual SLM microstructures experimentally
characterized. This section will describe the RVE generation, crystal
plasticity model that accounts for the cyclic deformation of grains, the FFT
homogenization framework, and the quantification methodology to predict
fatigue life based on the FIP.

###  3.1 RVE Generation

The digital representation of RVE corresponding to the SLM Hastelloy-X
microstructure is done to statistically include the grain size, shape, and
orientations obtained from the EBSD. The detailed RVE generation methodology
is enclosed in our previous work [56], and briefly recalled here for
completeness.

The 2D grain size distribution is experimentally obtained from the XY section
of EBSD images (see Fig 4) and is first converted to the apparent 3D grain
size, assuming spherical grains, using the software _GrainSizeTools_. [60].
The 3D distributions are then used as input to generate digital RVEs
consisting of equiaxial polycrystals. To this aim, a cloud of points and their
corresponding weights are generated to define a weighted Voronoi tessellation
which reproduced the targetgrain size distribution [59].

Finally, RVEs are scaled proportional to the aspect ratios measured. The
resulting synthetic RVEs are shown in Fig.5 with XY and XZ sections.

Figure 5:  3D images of RVEs of polycrystalline SLM Hastelloy-X for Bulk Z
(top left) and Bulk X (top right). Note that colors indicate the grain
numbers. EBSD maps of Bulk Z(bottom left) and Bulk X (bottom right) samples in
XY and XZ sections.The colors of the maps suggest the orientation of the
building direction compared to the reference frames following the IPF
triangle.

###  3.2 Crystal Plasticity Model

The crystal plasticity (CP) model developed by Cruzado et al.[61] for the
cyclic response of Inconel 718 is used in this study to simulate the elasto-
viscoplastic cyclic behavior of Hastelloy-X crystals. This model allows to
reproduce the isotropic and kinematic hardening and mean stress relaxation
effects which might appear under cyclic loading. The model selected is size
independent because the differences in grain sizes are below 10%, which would
have a negligible influence in the fatigue life. Moreover, kinematic hardening
is considered by a flow rule, in contrast to some strain gradient models used
in the context of fatigue [35] in which kinematic hardening is obtained as a
result of the computing explicitly the GNDs. The reason for this
simplification is having a computationally efficient model which allow us to
fulfil the two main requirements of our study (1) the use of realistic,
statistically representative and well discretized microstructures and (2)
considering fatigue life prediction in a statistical manner.

A multiplicative decomposition of the deformation gradient F into its elastic
($\text{{F}}^{e}$) and plastic ($\text{{F}}^{p}$) components is assumed,

$\textbf{F}=\text{{F}}^{e}\text{{F}}^{p}$ (1)

The plastic velocity gradient $\textbf{L}^{p}$ is computed in the intermediate
(relaxed) configuration as function of the shear rates
${\dot{\gamma}}^{\alpha}$ on all the slip systems $\alpha$,

$\textbf{L}^{p}=\dot{\textbf{{F}}}^{p}\textbf{F}^{p^{-1}}=\sum_{\alpha}\dot{\gamma}^{\alpha}\text{{s}}^{\alpha}\otimes\text{{m}}^{\alpha}$
(2)

where, $\text{{s}}^{\alpha}$ and $\text{{m}}^{\alpha}$ are the slip and slip
normal directions, respectively, in the initial configuration.

The second Piola-Kirchoff stress S depends linearly on the Green-Lagrange
strain $\textbf{E}^{e}$ through the elastic stiffness tensor $\mathbb{C}$,
which is given by

$\textbf{S}=\mathbb{C}\textbf{E}^{e}$ (3)

and the Cauchy stress tensor is calculated as,

$\sigma=\frac{1}{{J}^{e}}\textbf{F}^{e}\textbf{S}\textbf{F}^{e^{T}}$ (4)

, being ${J}^{e}$ the determinant of $\textbf{F}^{e}$.

A power-law is used to define the slip rate in each slip system as,

$\dot{\gamma}^{\alpha}=\dot{\gamma}_{0}\left(\frac{|\tau^{\alpha}-\chi^{\alpha}|}{{g}^{\alpha}}\right)^{\frac{1}{m}}\text{sign}(\tau^{\alpha}-\chi^{\alpha})$
(5)

where $\dot{\gamma}_{0}$ and $m$ are, respectively, the reference strain rate
and the strain rate sensitivity parameters. $\tau^{\alpha}$ is the resolved
shear stress, defined as,

$\tau^{\alpha}=\textbf{S}:(\text{{s}}^{\alpha}\otimes\text{{m}}^{\alpha}).$
(6)

The functions ${g}^{\alpha}$ and ${\chi}^{\alpha}$ are the critical resolved
shear stress and backstress on the $\alpha$ slip system respectively.

The evolution of the backstress on each slip system defines the kinematic
hardening. In the present model [61], a simplified version of the Ohno Wang
Model [62] limited to the first two terms is used,

$\dot{\chi}^{\alpha}=c\dot{\gamma}^{\alpha}-d\chi^{\alpha}|\dot{\gamma}^{\alpha}|\left(\frac{|\chi^{\alpha}|}{c/d}\right)^{k}$
(7)

where $c,d,$ and $k$ are material parameter. $k$ controls the mean stress
relaxation, while $c$ and $d$ stand for direct hardening and dynamic recovery
modulus, respectively.

The evolution of the Critical Resolved Shear Stress (CRSS) for a given slip
system, $g^{\alpha}$, defines the isotropic hardening and is given by

$\dot{g}^{\alpha}=\sum_{\beta}q_{\alpha\beta}h(\gamma_{a})|\dot{\gamma}^{\beta}|$
(8)

where $q_{\alpha\beta}$ are the latent hardening coefficients and $h$ is the
self hardening modulus which follows the Voce hardening model [63], given by

$h({\gamma_{a}})=h_{s}+\left[h_{0}-h_{s}+\frac{h_{0}h_{s}\gamma_{a}}{\tau_{s}-\tau_{0}}\right]exp^{\left(\frac{-h_{0}\gamma_{a}}{\tau_{s}-\tau_{0}}\right)}$
(9)

where, $\tau_{0},\tau_{s},h_{0}$ and $h_{s}$ are the hardening parameters and
the accumulated plastic shear $\gamma_{a}$ is computed as,

$\gamma_{a}=\sum_{\alpha}\int_{0}^{t}|\dot{\gamma}^{\alpha}|\mathrm{d}t.$ (10)

###  3.3 FFT based computational homogenization framework

The FFT homogenization code, FFTMAD [44, 59] is used to perform the virtual
fatigue test that provide the homogenized performance of polycrystals and the
localized stress and strain fields.

The polycrystal response is obtained by simulating the RVEs generated, which
are discretized into a regular array of voxels. The constitutive model
summarized in the previous section enters through a _UMAT_ subroutine [61].
The FFT homogenization problem in finite strains is solved using the Galerkin
FFT method [64, 65, 66]. The differential operators in the FFT solver used a
finite differentiation approach using the rotated scheme developed in [67].
Strain, stress and mixed control macroscopic boundary conditions are
introduced following the technique proposed by Lucarini et al. [44]. According
to this technique, the external macroscopic loading is controlled in terms of
macroscopic deformation gradient $\overline{F}_{ij}(t)$ or first Piola-
Kirchoff stress components $\overline{P}_{IJ}(t)$, for strain and stress-
controlled fatigue respectively [59]. The loading process is discretized in
increments and the resulting non-linear system of equations for each time
increment defines the deformation gradient which provides stress equilibrium.
The non-linear equation is solved using Newton Raphson [66] and the conjugate
gradient method is used as linear solver for each Newton iteration.

###  3.4 Fatigue Indicator Parameters and Life Prediction

The successive deformation of the polycrystal induces the localization of
plasticity in persistent slip bands which degenerate finally in microscopic
fatigue cracks [30]. The areas of the microstructure in which these bands and
subsequent cracks will appear correlate with regions where some Fatigue
Indicator Parameter (FIP) is maximum. Moreover, quantifying the value of this
FIP in that hot-spots allows to estimate the number of cycles to crack
nucleation. In the current study, the Fatigue Indicator Parameter through the
microstructure is obtained from computation homogenization of a full cycle.
The FIP distribution within the RVE was computed from the variation of micro-
fields and the state variables at every material point. This distribution
defines the hot-spots in which fatigue cracks are more likely to appear and
the fatigue life is predicted based on the most critical point. The most
common FIPs found in the literature are, accumulated plastic strain per cycle
[33, 30, 34], the plastic stored energy per cycle [35, 37], accumulated
internal strain energy and some other parameters.

In this work, plastic work per cycle $W_{cyc}(\textbf{x})$ is chosen as local
FIP, as it has shown to correlate well the fatigue life in many different
materials [38, 39, 40, 41]. This election is merely practical and other FIP
would probably lead to similar results. As an example, the stored energy
criterion [35, 37], would provide very close number of cycles, since the
stored energy is a small fraction of the total plastic work per cycle.

This FIP is defined as

$W_{cyc}(\textbf{x})=\int_{cyc}\tau_{\alpha}(\textbf{x})\dot{\gamma}_{\alpha}(\textbf{x})dt$
(11)

at each point x in the RVE being $\tau_{\alpha}$ and $\dot{\gamma}_{\alpha}$
the resolved shear stress and shear strain rate respectively. Eq.11 is applied
at the center of each voxel to obtain a local map of FIP values in the RVE.

However, the crack initiation on the persistent slip bands is a non-local
phenomenon, as shown for example in Castelluccio et al. [43]. For this reason,
and considering also the strong mesh dependence of the local FIP maps, non-
local FIP maps are defined as the volume averaged of local FIPs over the
grains or bands within a grain parallel to the slip planes with a constant
thickness [43, 40]. This type of non-local variable corresponds to an non-
local integral type model. Alternative non-local FIP by means of gradient
approaches can be derived, as proposed in [68]

The estimation of the number of cycles for crack nucleation is done based on
the most critical point of an RVE. In this work, the maximum of the non-local
FIPs defined at each band in the RVE is taken as driving force. More details
on the geometrical definition in bands can be found in [40]. Each grain is
divided in four families of bands, being $nb$ the total number of bands in the
RVE. The FIP of the full RVE is then defined as

$W_{cyc}^{b}=\max_{i=1,nb}\left\\{\max_{\beta_{i}}\frac{1}{V_{i}}\int_{V_{i}}W_{cyc}^{\beta_{i}}(\mathbf{x})dV_{i}\right\\}$
(12)

where $1\leq i\leq nb$ refers to each of the bands in the RVE,
$\beta_{i}=1,2,3$, corresponds to the three different slips systems contained
in the slip plane parallel to that band and $V_{i}$ is the volume of the band.

Normally, crack nucleation is directly related to the cyclic FIP obtained from
Eq.12 using some empirical relations. However, in this work, a different
methodology has been used to predict both fatigue life for stress- and strain-
controlled loading, keeping a similar definition of FIP.

#### Strain Controlled Loading

The value of the FIP in a stable cycle can be used to estimate, based on
extrapolation, the number of cycles for crack nucleation. The relation between
this value and the number of cycles for nucleation can be based on a linear
extrapolation [33] or on other relation, as the power law in [40, 41]. In the
latter case, the value of the FIP in a stable cycle, $W_{cyc}$, is linked to
the number of cycles for nucleation $N_{i}$ as

$N_{i}=\frac{W_{crit}}{{(W_{cyc})}^{m}},$ (13)

where $W_{crit}$ and $m$ are two parameters of the material. This biparametric
fatigue life prediction law requires the use of two independent fatigue
experiments [40] to be calibrated.

The use of the previous approach implies direct simulation of the RVE
deformation until a stable hysteresis cycle is reached. Under strain control,
hysteresis evolves from cycle to cycle, and stabilization can be reached after
a few cycles in some cases to thousands of cycles in other cases, depending on
the material and loading conditions [33]. To limit the computational cost of
these simulations, different strategies have been developed in the context of
polycrystalline homogenization, such as the wavelet transformation-based
multi-time scaling algorithm (WATMUS) [69] or the linear extrapolation of
internal variables or cycle jumps [61].

#### Stress Controlled Loading

Contrary to strain-controlled tests, under stress-control loading, the plastic
flow is concentrated mainly in the first few cycles. From a computational
point of view, this usually results in a relatively fast transition to an
approximate elastic cyclic macroscopic response. Although microplasticity can
still be observed after the shakedown regime, the associated FIPs are three to
four orders of magnitude lower than the FIPs in plastic regimes, too small for
an accurate prediction of crack nucleation based on the extrapolation
approaches used for strain control.

As an alternative, we consider that the total accumulated plastic deformation
(also related to the stored energy) until the shakedown limit is the precursor
to the nucleation of a crack. Therefore, in this work, the total accumulated
plastic deformation, $W^{acc}_{p}$, is proposed as FIP to correlate the number
of cycles for crack nucleation and the microplasticity effects after shaking
are neglected in this correlation.

The approach proposed here first introduces a criterion to quantify, based on
microscopic fields in an RVE simulation, the number of cycles $n=ns$ at which
elastic shakedown occurs.

$ns\ \text{such}\ \text{that}\ W_{cyc}(n=ns)=10^{-4}W_{cyc}(n=1).$ (14)

where $W_{cyc}(n)$ is the FIP value in the RVE (Eq. 12 ) on the $n^{th}$
cycle. Then, the accumulated plastic deformation, $W^{acc}_{p}$, is given as

$W^{acc}=\sum_{n=1}^{n=ns}W_{cyc}(n)$ (15)

A power-law function is proposed (Eq. 16) which also includes two fitting
parameters, $W^{crit}_{\sigma}$, and $nk$ to link the accumulated plastic FIP,
$W^{acc}_{p}$, and the fatigue crack initiation life $N_{i}$.

$N_{i}=\frac{W^{crit}_{\sigma}}{{(W^{acc}_{p})}^{nk}}$ (16)

##  4 Results

###  4.1 Experimental results

Uniaxial tensile fatigue tests were performed using strain control in the case
of Bulk X samples and under stress control for both Bulk X and Bulk Z. The
cylindrical-shaped bulk sample (Bulk Z) was subjected to uniaxial tensile
cyclic loading in the building direction (Z), while the rectangular-shaped
sample ( Bulk X) was subjected to uniaxial tensile cyclic loading in the
X-direction. In the case of strain controlled tests, the hysteresis stress-
strain cycle was registered cycle by cycle in order to obtain the cyclic
plastic response. From now on, due to the confidentiality agreement signed
with the industrial partner, all data are normalized by constant normalization
factors. The normalization factor $\Delta\epsilon_{min}$ corresponds to the
minimum applied strain range under strain loading conditions. For stress, the
normalization factor is the critical resolved shear stress, $\tau_{0}$,
identified for Hastelloy-X grains at $750^{\circ}$C.

The cyclic uniaxial tests under strain control were carried out with strain
amplitudes, $\frac{\Delta\epsilon}{\Delta\epsilon_{min}}$= 1, 1.1, 1.2, 1.3,
1.4, 1.5, 1.6, 1.7 and, 1.8 under strain ratio $R_{\epsilon}$=0. The
hysteresis cycles show a combination of isotropic and kinematic hardening and
finally collapse in a stable cycle. An example of these experimental results
is represented in Fig. 6, where the stress-strain cycle for the first and
$N=2000$ cycles (when the cycle became stable) is represented for two strain
ranges, $\frac{\Delta\epsilon}{\Delta\epsilon_{min}}$= 1 (Fig.6(a) and
$\frac{\Delta\epsilon}{\Delta\epsilon_{min}}$= 1.6 (Fig.6(b)). The rest of the
results will be represented together with the model predictions in the next
section.

In the case of stress-controlled tests, the applied uniaxial stress range
considered was $\frac{\Delta\sigma}{\tau_{0}}$= 3.2, 3.57, 3.93, 4.29, 4.64,
and 5 under stress ratio $R_{\sigma}$=0.03. Both stress and strain controlled
tests were carried out at 750°C using a trapezoidal (1s-1s-1s-1s) waveform
with a frequency of 0.25Hz and a stress concentration factor ($K_{t}=1$).

(a)

(b)

Figure 6: _Under strain controlled loading, the stress-strain cycle for the
first(shown in red color), and $N=2000$ cycles(shown in blue color) are
represented for two strain ranges, (a)
${\Delta\epsilon}/{\Delta\epsilon_{min}}$= 1, and (b)
${\Delta\epsilon}/{\Delta\epsilon_{min}}$= 1.6 _

The fatigue $\varepsilon$-N curve obtained for strain control and S-N curve
for stress control are represented in Fig.7(a) and Fig. 7(b), respectively. In
these curves, the fatigue life $N$ corresponds to the cycle in which the final
fracture is reached on the specimen. The fatigue performance of the samples
built in the X and Z directions 7(b) is different, especially for high stress
levels where Bulk X samples present longer fatigue life. This result is in
agreement with previous studies by Wang et al.[26] where the better
performance of the samples built in the X direction is also observed for high
stresses.

(a)

(b)

Figure 7: _The experimental fatigue (a)strain-life curve obtained using strain
control loading for Bulk X and (b)stress-life curve obtained using stress
control loading for Bulk X and Bulk Z samples._

To observe the mechanism of fatigue failure, a fractography analysis was
performed for the two samples loaded with cyclic strain amplitude
${\Delta\epsilon}/{\Delta\epsilon_{min}}$, 1 and 1.6 at $750^{\circ}$C. The
fracture surfaces obtained are shown in Fig.8(a), and Fig.8(b) respectively,
showing the crack initiation (flat areas origin of propagating cracks),
propagation (beach marks), and final fracture regions (dimples). It is
observed that the dominant crack initiation sites were subsurface areas that
have some small defects (both unmelted particles or very small pores have been
observed), as they act as stress concentration sites. For low strain amplitude
${\Delta\epsilon}/{\Delta\epsilon_{min}}$=1, multiple crack initiation sites
were observed, while at higher applied
strains,${\Delta\epsilon}/{\Delta\epsilon_{min}}$=1.6, a single and well-
defined crack initiation site was observed.

Multiple crack initiation sites for small strain amplitude
(${\Delta\epsilon}/{\Delta\epsilon_{min}}$=1) implies that very small
heterogeneities near the surface are enough to nucleate a crack and therefore
the expected scatter in these cases is greater.

Figure 8:  An overall view of the specimen subjected to cyclic strain
amplitude, ${\Delta\epsilon}/{\Delta\epsilon_{min}}$= a) 1 and b) 1.6, $\%s$
showing crack initiation, propagation, and final fracture regions.

The number of cycles for propagation can be qualitatively estimated by the
number of beach marks observed in the propagation area. The crack propagation
area for both strain ranges is similar, but since the width of the beach marks
is smaller for the lower strain amplitude, the propagation regime is slightly
larger for the smaller strain range. A rough estimation of the number of
propagation cycles is around $0.01N_{f}$ for
${\Delta\epsilon}/{\Delta\epsilon_{min}}$ = 1 and $0.1N_{f}$ for
${\Delta\epsilon}/{\Delta\epsilon_{min}}$=1.6, where $N_{f}$ is the
experimental fatigue life. In both cases, the ratio of propagation to total
life is small and nucleation can be considered as the most dominant fatigue
mechanism for both applied strains. This fact validates the modeling approach
followed, in which fatigue life is estimated based on the number of cycles for
nucleation.

The final fracture of the samples was reached when the crack reached the
critical size, and the fracture surface area corresponding to this stage was
easily observed as a typical ductile fracture surface with dimples ( Fig.9).

Figure 9:  The final fracture surface of samples subjected to cyclic strain
amplitude, ${\Delta\epsilon}/{\Delta\epsilon_{min}}$= a) 1 and b) 1.6, $\%s$
showing ductile failure with dimples.

###  4.2 Numerical results

The CP-FFT framework presented is applied to predict the cyclic response of
bulk SLM Hastelloy-X built in different orientations, thus presenting
different microstructural characteristics.

The first objective is to obtain the crystal plasticity parameters for an
accurate prediction of the cyclic response for all of the specimens
considered. Second, the fatigue life prediction model will be adjusted based
on two fatigue tests under strain control. The resulting fatigue life
prediction tool will be validated for the rest of the strain-controlled
fatigue tests and also for the stress-controlled loading. Finally, the effect
of the microstructures resulting from the different SLM processes on fatigue
performance will be analyzed.

### Crystal plasticity model parameters

The parameters of the CP model of SLM Hastelloy-X at $750^{\circ}$C after
annealing and thermal processing will be obtained. The elastic constants of
Hastelloy-X single crystals are taken from Canistraro et al.[70]. The strain
rate sensitivity coefficient and exponent, $\dot{\varepsilon}_{0}$ and $m$,
respectively, are taken from previous work in Hastelloy-X [56]. The parameters
defining the critical resolved shear stress and the evolution of the back
stress, which are responsible for isotropic and kinematic hardening,
respectively, are obtained by inverse fitting. The input data is the
experimental uniaxial cyclic stress-strain response of the bulk sample X
loaded in the X direction with strain control and
${\Delta\epsilon}/{\Delta\epsilon_{min}}$= 1.6, the one with the highest
strain amplitude. The RVE used for the identification of CP parameters is
generated to be statistically representative of the actual microstructure
(same grain size, aspect ratio, and orientation distributions). The RVE
accommodates approximately 1000 grains and is discretized in 643 voxels. The
inverse optimization framework proposed by Herrrera-Solaz et al.[71], based on
the Levenberg-Marqadt least squares algorithm, is used in this study.

Experimentally, it is observed that for SLM-Hastelloy-X, the cyclic stress-
strain behavior stabilizes approximately after 2000 cycles for strain-
controlled loading at $750^{\circ}$C. Numerically, simulating 2000 cycles
makes the optimization process computationally too expensive. Some
acceleration techniques have been proposed in the literature, such as the
cycle jump approach [61]. However, since the microstructure-sensitive fatigue
crack initiation methodology is based on extrapolating the FIP distribution in
the stable hysteresis loop, in this work we have opted for a simpler
alternative. In our approach, the CP parameters controlling isotropic and
kinematic hardening evolution have been fitted such that 15 numerical cycles
of the RVE provide a stable hysteresis cycle equal to the experimental stable
cycle, reached after 2000 cycles. The number of 15 cycles has been chosen
large enough to ensure a progressive stabilization of the cyclic response and
local history fields and FIPs.

The CP parameters correspond to the Voce isotropic hardening
($\tau_{0},\tau_{s}$, $h_{0}$ and $h_{s}$), and the kinematic hardening of the
Ohno-Wang model ($c,d,k$) are identified using the procedure mentioned above,
and these values are tabulated in Table 2 along with the remaining CP
parameters. The stable experimental cyclic stress-strain curve of the 2000th
cycle is represented in Figure 10 together with the polycrystalline cyclic
response of the 15th cycle after adjustment of the CP parameters. The figure
shows an almost superimposed response between the model and the experiment,
confirming the accuracy of fitting the parameters of the CP model.

Table 2: Crystal plasticity parameters of a SLM Hastelloy-X at $\mathbf{750^{\circ}}$C Elastic $(GPa)$ | $\text{C}_{11}$ | $\text{C}_{12}$ | $\text{C}_{44}$ |
---|---|---|---|---
| 194.8 | 126.91 | 89.95 |
Viscoplastic | $m$ | ${\dot{\gamma}}_{0}(s^{-1})$ |  |
| 0.017 | $\text{2.42}\times{\text{10}}^{-3}$ |  |
Isotropic Hardening (MPa) | $\bm{\tau_{0}}$ | $\bm{\tau_{s}}$ | $\bm{h_{0}}$ | $\bm{h_{s}}$
| $\bm{\tau_{0}}$ | $\bm{\textbf{1.25}\tau_{0}}$ | $\textbf{78.7}\bm{\tau_{0}}$ | $\textbf{0.71}\bm{\tau_{0}}$
Kinematic Hardening |  c(MPa) | $\bm{d}$ | $\bm{k}$ |
| $\mathbf{171.42\tau_{0}}$ | $\mathbf{2.57\tau_{0}}$ | $\mathbf{0.1066}$ |
Figure 10:  A stable cyclic experimental loop for applied strain range,
${\Delta\epsilon}/{\Delta\epsilon_{min}}$=1.6 and numerically fitted cyclic
loop using LM algorithm.

###  4.3 Prediction of cyclic plastic response

The ability of the set of CP parameters fitted above to predict the cyclic
deformation behavior for different applied strain amplitudes is presented in
this section. In particular, the stress-strain curves and the evolution of
minimum and maximum stress with the number of real cycles are predicted under
the uniaxial cyclic strain amplitude
$\frac{\Delta\epsilon}{\Delta\epsilon_{min}}$= 1, 1.1, 1.2, 1.4, 1.6, 1.8,
with $R_{\epsilon}=0$. All the cases considered correspond to bulk-X samples.

The stable experimental hysteresis curve, corresponding to the 2000th
experimental cycle, is represented in Fig. 11 for six different strain
amplitudes together with the computational homogenization results for the 15th
cycle. The comparison shows that an accurate stabilized cyclic stress-strain
loop can be predicted for all the applied strain amplitudes. These results
validate the simulation strategy, based on accelerating the hardening
evolution, to predict the stable cycle under a wide range of cyclic loading
conditions.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 11: _Comparison between stable experimental 2000th cycle (solid red
lines), and numerical cyclic 15th cycle (dotted blue lines) hysteresis loops
for different applied strain ranges,(
${\Delta\epsilon}/{\Delta\epsilon_{min}}$= a)1, b)1.1, c)1.2, d)1.4, e)1.6,
and f)1.8. Note that $\tau_{0}$ normalizes stresses. _

The evolution of peak stresses until cycle stabilization is represented in
Fig.12 for both the experimental results and the numerical predictions. The
experiments show an initial asymmetry in tension/compression maximum stresses
that progressively disappears and that is also captured by the simulations. It
can be observed that the experimental results always stabilized before 2000
cycles and the numerical value for 15 cycles coincides with the stable value
for all the strain range. The evolution of the stress peaks obtained by the
model before stabilization also follows the experimental trend. The comparison
of experimental and predicted peak stresses in Fig.12 show a considerably
larger difference (up to $20\%$) in the initial cycles with respect the rest
of the cycles. The reason is that for large applied strains a fast decay in
the maximum stresses is observed in the experiments for a small number of
cycles, and the proposed acceleration, where each model cycle corresponds to
125 cycles, cannot reproduce that decay preserving the accuracy for the more
gradual cases. Nevertheless, the difference between peak stresses at the 15th
numerical cycle and the 2000th experimental cycle for tensile loading is
always less than $6\%$ and less than $8\%$ for compressive loading, even for
the large applied strain cases where the initial decay was spread in more
cycles. This small differences in all the very different cases considered show
a remarkable accuracy of the model in predicting stabilized cyclic response
with the accelerated technique.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 12: _Comparison between experimental(solid red lines) and numerical
cyclic(dotted blue lines) response based on the evolution of maximum and
minimum stresses with the number of cycles for different applied strain
ranges,( ${\Delta\epsilon}/{\Delta\epsilon_{min}}$= a)1, b)1.1, c)1.2, d)1.4,
e)1.6, and f)1.8. Note that $\tau_{0}$ normalizes stresses. _

###  4.4 Fatigue Life Prediction

####  4.4.1 Life prediction model calibration

The parameters of the fatigue life models, $W_{crit}$ and $m$ in Eq.13 and
$W^{crit}_{\sigma}$ and $n$ in Eq.15, respectively, for the strain- and
stress-controlled loading, must be calibrated using some of the experimental
data. In each case, two different experimental results and their corresponding
cyclic behavior are necessary to predict the power law constants in Eqs.13 and
15. Furthermore, to consider the stochastic nature of fatigue, the average
response of the simulation of 20 different RVE realizations (also known as
Statistical Volume Element, SVE) was considered for each applied load. The use
of an SVE is necessary to consider the differences between the results of
different RVEs statistically. The fatigue model parameters obtained for the
strain-controlled loading and stress-controlled loading are tabulated in Table
3.

Table 3: The fitted parameters for the proposed fatigue crack initiation models  |  $\varepsilon$ \- controlled loading |  $\sigma$ \- controlled loading
---|---|---
Parameter | $W_{crit}$ | $m$ | ${W_{\sigma}}^{crit}$ | $nk$
Value | $3.421\times 10^{19}$ | $2.62$ | $4.06\times 10^{8}$ | $-0.56$

###  4.5 Fatigue Life Prediction: strain controlled tests

Figure 13:  Experimental fatigue life and model predictions for bulk X under
strain-controlled loading at $750^{\circ}$C. Note that the minimum cyclic
strain amplitude $\Delta\epsilon_{min}$ normalizes strain.

.

The cyclic stable FIPs and fatigue model in Eq. 13 are used to predict fatigue
life as a function of applied strain and microstructure. The experimental
fatigue lives obtained for different experiments at each strain amplitude are
plotted in Fig. 13 using red stars. The average experimental results for two
strain ranges $\frac{\Delta\epsilon}{\Delta\epsilon_{min}}$= 1 and 1.8, marked
with black dots, and the average FIP of the simulation of 20 RVEs for these
two applied strain ranges are used to calibrate the fatigue model parameters.
Figure 13 represents, using small blue curves near each experimental point,
the fatigue life probability distribution predicted by using the results of
the 20 RVEs, for each strain amplitude. Very good agreement between the
experiments and simulations is observed for all applied cyclic strain
amplitudes. When comparing the average experimental fatigue life with the peak
of probability of failure provided by the model distributions, the errors are
below 20% except for the case of
$\frac{\Delta\epsilon}{\Delta\epsilon_{min}}$= 1.3, where the error grows to
61%. Moreover, the average experimental life lied in all the cases in the
percentile 95% of the model predictions, with the exception of the 1.3 case.
The experimental life for this case breaks the tendency observed in the
experimental results, and is longer than for a lower strain range of 1.2.
Model cannot capture this abnormal tendency that is probably caused by some
experimental issue.

Furthermore, the results highlighted the predictive capacity of the model to
consider the change in the FIP distribution between high and low strain
amplitudes. The distribution of plastic deformation is heterogeneous at lower
strain amplitudes, and very few grains undergo plastic deformation. However,
at higher strain amplitudes, more homogeneous plastic deformation was
observed. The effect of microstructure has a more pronounced effect at lower
strain amplitudes than at higher strain amplitudes, since the localized
continuous accumulation of plastic slip in very few grains leads to
considerable experimental and numerical scatter in the fatigue life.

###  4.6 Fatigue Life Prediction: Stress Controlled tests

The crystal plasticity model presented in section 3.2 is extended to predict
fatigue life under stress-controlled loading. In this case, tests are
available in two different building directions, Bulk X and Bulk Z, so the
predictive capacity of the model for different microstructures will be
assessed.

The methodology to predict fatigue life under stress-controlled loading is
described in Section 3.4. Cumulative cyclic damage in the plastic regime
($W_{p}^{acc}$) is specified as the precursor to the onset of fatigue cracks,
and fatigue live $N$ is predicted using the modified criteria (Eq.15-16).
Since under stress control, the scatter is usually larger, 50 RVEs were
generated with an average grain size, aspect ratio, and grain orientations for
Bulk X and Bulk Z samples. Virtual uniaxial cyclic tests were performed on
RVEs under stress-controlled loading with ${\Delta\sigma}/{\tau_{0}}$= 3.2,
3.57, 3.93, 4.29, 4.64, and 5 and under a stress ratio $R_{\sigma}$ = 0.03.
Note that $\tau_{0}$ corresponds to the critical resolved shear stress(CRSS).
The adjustment parameters $W^{crit}_{\sigma}$ and $nk$ in Eq.16, which are
necessary to predict the fatigue crack initiation under stress-controlled
loading, were identified from the fatigue life obtained in two different tests
(${\Delta\sigma}/{\tau_{0}}=3.2,5$) for a unique type of material, the one
corresponding to fabrication in the X direction (Bulk X). The values of the
fitting parameters are $W^{crit}_{\sigma}=4.06\times 10^{8}$ and $nk=-0.56$.
The accumulated FIP, $W_{p}^{acc}$ of these tests was obtained from the
simulation of a SVE with 50 different RVEs. The large size of the SVE helps
predict the stochastic nature of the fatigue life for a given microstructure
and loading conditions.

Once the life prediction model is fitted, the SVE for Bulk-X samples is
simulated and used to obtain the fatigue life prediction for the rest of the
cases. In the case of Bulk-Z, a new SVE is generated containing 50
statistically representative RVEs of the microstructure of the material, which
is different from the one of the Bulk-X samples. This SVE is then used to
predict the fatigue life of these samples. It is important to note that no
experimental results on Bulk-Z material have been used to calibrate the model,
so the results are pure predictions and will assess the validity of the model
to consider different microstructures for the same parameters.

The fatigue crack initiation predictions of the model for different applied
cyclic stress ranges, ${\Delta\sigma}/{\tau_{0}}$ are represented in Fig. 14
for the orientation of the sample, (a) Bulk X and (b) Bulk Z, together with
the respective experimental life.

(a)

(b)

Figure 14: _Experimental results and numerical model predictions fatigue crack
initiation life for SLM Hastelloy-X at $750^{\circ}$C as a function of cyclic
applied stress,${\sigma}/{\tau_{0}}$. Note that the stress is normalized by
$\tau_{0}$. a) Bulk X b) Bulk Z. These results are for 50 RVE realizations for
each applied stress, with a confidence level of $95\%$. _

The agreement between experimental and numerically predicted fatigue life is
very accurate for both building directions in most of the strain ranges
studied. The largest difference was found for the bulk Z sample in the stress
ranges ${\Delta\sigma}/{\tau_{0}}$, 3.57 and 3.93, where fatigue life is
slightly underestimated. These results highlight that the model can accurately
predict the fatigue life that accounts for the effect of the microstructure.
The agreement with the experimental results for the Bulk-Z direction is
remarkable because the fatigue life law used for the predictions was
calibrated only using two experiments of Bulk-X samples.

To analyze the effect of the building direction, the S-N curves for the two
building directions together are represented in Fig. 15. In this figure, it
can be observed that the direction of SLM fabrication significantly affects
the fatigue life of Hastelloy-X. Experimentally, the horizontally built
specimen (Bulk X) shows superior resistance to the initiation of fatigue
cracks compared to the vertically built specimen (Bulk Z), with the exception
of the two smallest stress ranges applied. These results are in agreement with
the observations of Wang et. al. [26] for the same alloy, Hastelloy-X, and of
Bayati et al.[72] for SLM Ni-Ti. Furthermore, the better fatigue performance
of the printed Bulk-X Hastelloy-X samples corresponds to a stronger
quasistatic response, as shown in [56].

The model can reproduce the better performance of Bulk X compared to Bulk Z
for stresses greater than 3.93$\tau_{0}$ observed in the experiments. Multiple
factors could potentially contribute to this anisotropic fatigue performance,
such as porosity, residual stresses, surface roughness, and orientation of the
deposited layer with respect to the loading direction. However, the
fabrication was designed to minimize most of them. For example, minimal
porosity was observed in the samples as a result of an optimal selection of
the SLM parameters, and a fine machining ensures that the sample has very
minimal surface roughness. With respect to the level of residual stresses, it
should be very small due to the annealing process and heat treatment.
Therefore, the differences observed in the experiments should be mainly caused
by the different microstructures (grain size, aspect ratio, and orientation)
generated during SLM fabrication in different directions with respect to the
loading axis. In the absence of porosity and defects, the slip band precursors
of the nucleating cracks tend to appear in grains, whose shape, orientation
with respect to macroscopic load, and surrounding area are specially suited
for localizing plastic deformation and, therefore, the polycrystalline
microstructure plays an essential role. Since the two materials have a very
similar texture and average grain size, the main difference in their
microstructures is their grain shape, having Bulk X grains a lower aspect
ratio. The more equiaxed grain shape should then provide better resistance to
fatigue, because the slip bands are larger and localize more plastic strain in
the elongated grains.

Both the model and the experiments agree that the effect of anisotropy in
fatigue life increases with the applied stress, being maximal for the highest
stress amplitudes, where most of the grains show plastic deformation. At lower
stresses, for long LCF lifes, the model shows a similar fatigue endurance for
both samples. The prediction of a reduction in fatigue performance anisotropy
has also been observed for SLM samples [26, 72].

Figure 15:  Comparison between predicted and experimental fatigue lives of
Bulk X and Bulk Z, showing the effect of build orientation on fatigue lives.

In summary, in addition to the effect of defects and porosity, the orientation
of the building influences the lifetime of fatigue. The S-N curve shows a
higher slope for horizontally oriented samples and lower slopes for vertically
oriented samples. The reason behind the effect of the building direction on
fatigue performance is the difference in the microstructures developed during
the process. This influence is strong at higher applied stresses; however,
such effects are minimal at lower applied stresses. Therefore, to design the
SLM of components against fatigue loading, microstructural aspects (grain
aspect ratio, average size, and orientation result of the building orientation
should be considered, as they significantly influence fatigue behavior.
Furthermore, to capture this fatigue anisotropy, the use of microstructure-
sensitive fatigue life nucleation approaches based on polycrystalline
homogenization is essential.

##  5 Conclusion

A microstructure sensitive fatigue life prediction framework, based on CP-FFT,
is proposed to model the fatigue performance of SLM fabricated Hastelloy-X at
$750^{\circ}$C considering the effect of the microstructures developed during
the SLM process. The strategy is based on obtaining probability distributions
of fatigue indicator parameters (FIP) from the simulation of the cyclic
response of a statistical volume element (SVE) formed by several
representative volume elements (RVE). Then, the fatigue life is obtained for
each particular case from the FIP distribution obtained through an empirical
law calibrated using two fatigue experiments. The microstructure enters the
model by statistical representation in the RVE of the shape, size, and
orientation distribution of the grains, which are generated from experimental
data. The strategy allows fatigue life prediction for both strain- and stress-
controlled experiments by using different FIPs, based on cyclic and total
dissipated work, respectively. The framework has been applied to Hastelloy-X
built in different longitudinal directions. The main conclusions about the
modeling framework and its application to Hastelloy-X are the following.

  * 1.

A phenomenological crystal plasticity model that includes a back stress term
can accurately reproduce the Hastelloy-X hysteresis curve under all loading
conditions.

  * 2.

Accelerating the evolution of internal variables so that the stable hysteresis
cycle of the model is reached before experimental stabilization strongly
reduces computational cost and provides an accurate result in fatigue
prediction.

  * 3.

A power-law life prediction model with two fitting parameters, based on cyclic
and accumulated plastic work FIP, is able to accurately predict life for
strain and stress controlled tests, respectively.

  * 4.

The direction of the SLM fabrication (along the loading axis or perpendicular
to it) is found to influence the fatigue performance of Hastelloy-X,
especially for high applied stress ranges.

  * 5.

The model can capture the effect of the fabrication direction on fatigue
performance by using RVEs statistically representing the actual microstructure
resulting from each fabrication direction.

  * 6.

The model shows that grain elongation is the most relevant microstructural
feature responsible for the better fatigue performance of Bulk-X samples.

## References

  * [1] M.A.Rowley, E.A.Thornton, Constitutive modeling of the visco-plastic response of Hastelloy-X and Aluminum Alloy 8009, Journal of Engineering Materials and Technology 118 (19–27) (1996).
  * [2] M. M. Attallah, R. Jennings, X. Wang, L. N. Carter, Additive manufacturing of Ni-based superalloys: The outstanding issues, MRS Bulletin 41 (2016) 758–764.
  * [3] M. L. Montero-Sistiaga, S. Pourbabak, J. Van Humbeeck, D. Schryvers, K. Vanmeensel, Microstructure and mechanical properties of Hastelloy X produced by HP-SLM (high power selective laser melting), Materials and Design 165 (2019) 107598.
  * [4] D. Barba Cancho, C. Alabort Martínez, T. Tang, M. Viscasillas, R. Reed, E. Alabort, On the size and orientation effect in additive manufactured Ti-6Al-4V, Materials and Design 186 (2019) 108325.
  * [5] J. J. Lewandowski, M. Seifi, Metal additive manufacturing: A review of mechanical properties, Annual Review of Materials Research 46 (2016) 151–186.
  * [6] A. E. Wilson-Heid, Z. Wang, B. McCornac, A. M. Beese, Quantitative relationship between anisotropic strain to failure and grain morphology in additively manufactured ti-6al-4v, Materials Science and Engineering: A 706 (2017) 287–294.
  * [7] C. Pei, D. Shi, H. Yuan, H. Li, Assessment of mechanical properties and fatigue performance of a selective laser melted nickel-base superalloy inconel 718, Materials Science and Engineering: A 759 (2019) 278–287.
  * [8] Y. Kok, X. Tan, P. Wang, M. Nai, N. Loh, E. Liu, S. Tor, Anisotropy and heterogeneity of microstructure and mechanical properties in metal additive manufacturing: A critical review, Materials and Design 139 (2018) 565–586.
  * [9] M. Aydinöz, F. Brenne, M. Schaper, C. Schaak, W. Tillmann, J. Nellesen, T. Niendorf, On the microstructural and mechanical properties of post-treated additively manufactured inconel 718 superalloy under quasi-static and cyclic loading, Materials Science and Engineering: A 669 (2016) 246–258.
  * [10] M. M. Kirka, D. A. Greeley, C. Hawkins, R. R. Dehoff, Effect of anisotropy and texture on the low cycle fatigue behavior of inconel 718 processed via electron beam melting, International Journal of Fatigue 105 (2017) 235–243.
  * [11] P. Kanagarajah, F. Brenne, T. Niendorf, H. Maier, Inconel 939 processed by selective laser melting: Effect of microstructure and temperature on the mechanical properties under static and cyclic loading, Materials Science and Engineering: A 588 (2013) 188–195.
  * [12] P. Edwards, M. Ramulu, Fatigue performance evaluation of selective laser melted ti–6al–4v, Materials Science and Engineering: A 598 (2014) 327–337.
  * [13] S. Gribbin, J. Bicknell, L. Jorgensen, I. Tsukrov, M. Knezevic, Low cycle fatigue behavior of direct metal laser sintered inconel alloy 718, International Journal of Fatigue 93 (2016) 156–167.
  * [14] A. S. Johnson, S. Shao, N. Shamsaei, S. M. Thompson, L. Bian, Microstructure, fatigue behavior, and failure mechanisms of direct laser-deposited inconel 718, Jom 69 (3) (2017) 597–603.
  * [15] D. Agius, K. I. Kourousis, C. Wallbrink, T. Song, Cyclic plasticity and microstructure of as-built SLM ti-6al-4v: The effect of build orientation, Materials Science and Engineering: A 701 (2017) 85–100.
  * [16] P. F. Kelley Jr, Fatigue behavior of direct metal laser sintered (dmls) inconel 718, Ph.D. thesis, Tufts University (2016).
  * [17] W. Li, R. Sun, T. Hu, X. Li, C. Li, Y. Zhang, X. Ding, P. Wang, Effect of elevated temperature on high-cycle and very-high-cycle fatigue properties of ni-based superalloy manufactured by selective laser melting, International Journal of Fatigue 148 (2021) 106250.
  * [18] R. Konečná, G. Nicoletto, L. Kunz, A. Bača, Microstructure and directional fatigue behavior of inconel 718 produced by selective laser melting, Procedia Structural Integrity 2 (2016) 2381–2388.
  * [19] A. Yadollahi, N. Shamsaei, S. M. Thompson, A. Elwany, L. Bian, Effects of building orientation and heat treatment on fatigue behavior of selective laser melted 17-4 PH stainless steel, International Journal of Fatigue 94 (2017) 218–235.
  * [20] R. Esmaeilizadeh, A. Keshavarzkermani, S. Faghih, B. Behravesh, U. Ali, A. Bonakdar, H. Jahed, E. Toyserkani, Fatigue characterization and modeling of additively manufactured hastelloy-x superalloy, Journal of Materials Engineering and Performance (Feb. 2022).
  * [21] S. Leuders, M. Thöne, A. Riemer, T. Niendorf, T. Tröster, H. Richard, H. Maier, On the mechanical behaviour of titanium alloy TiAl6v4 manufactured by selective laser melting: Fatigue resistance and crack growth performance, International Journal of Fatigue 48 (2013) 300–307.
  * [22] J. Strößner, M. Terock, U. Glatzel, Mechanical and microstructural investigation of nickel-based superalloy IN718 manufactured by selective laser melting (SLM), Advanced Engineering Materials 17 (8) (2015) 1099–1105.
  * [23] T. Lindström, D. Ewest, K. Simonsson, R. Eriksson, J.-E. Lundgren, D. Leidermark, Constitutive model of an additively manufactured ductile nickel-based superalloy undergoing cyclic plasticity, International Journal of Plasticity 132 (2020) 102752.
  * [24] Q. Han, R. Mertens, M. L. Montero-Sistiaga, S. Yang, R. Setchi, K. Vanmeensel, B. V. Hooreweder, S. L. Evans, H. Fan, Laser powder bed fusion of hastelloy x: Effects of hot isostatic pressing and the hot cracking mechanism, Materials Science and Engineering: A 732 (2018) 228–239.
  * [25] R. Esmaeilizadeh, A. Keshavarzkermani, U. Ali, B. Behravesh, A. Bonakdar, H. Jahed, E. Toyserkani, On the effect of laser powder-bed fusion process parameters on quasi-static and fatigue behaviour of hastelloy x: A microstructure/defect interaction study, Additive Manufacturing 38 (2021) 101805\.
  * [26] F. Wang, Mechanical property study on rapid additive layer manufacture hastelloy® x alloy by selective laser melting technology, The International Journal of Advanced Manufacturing Technology 58 (5-8) (2011) 545–551.
  * [27] S. Lee, J. W. Pegues, N. Shamsaei, Fatigue behavior and modeling for additive manufactured 304l stainless steel: The effect of surface roughness, International Journal of Fatigue 141 (2020) 105856.
  * [28] S. Romano, L. Patriarca, S. Foletti, S. Beretta, Lcf behaviour and a comprehensive life prediction model for alsi10mg obtained by slm, International Journal of fatigue 117 (2018) 47–62.
  * [29] R. Branco, J. Costa, F. Berto, S. Razavi, J. Ferreira, C. Capela, L. Santos, F. Antunes, Low-cycle fatigue behaviour of AISI 18ni300 maraging steel produced by selective laser melting, Metals 8 (1) (2018) 32.
  * [30] D. McDowell, F. Dunne, Microstructure-sensitive computational modeling of fatigue crack formation, International Journal of Fatigue 32 (9) (2010) 1521–1542.
  * [31] J. Segurado, R. A. Lebensohn, J. LLorca, Computational homogenization of polycrystals, in: Advances in Applied Mechanics, Elsevier, 2018, pp. 1–114.
  * [32] S. Lucarini, J. Segurado, An upscaling approach for micromechanics based fatigue: from rves to specimens and component life prediction, International Journal of Fracture 223 (1) (2020) 93–108.
  * [33] A. Manonukul, F. P. E. Dunne, High– and low–cycle fatigue crack initiation using polycrystal plasticity, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 460 (2047) (2004) 1881–1903.
  * [34] C. Sweeney, B. O’Brien, F. Dunne, P. McHugh, S. Leen, Strain-gradient modelling of grain size effects on fatigue of CoCr alloy, Acta Materialia 78 (2014) 341–353.
  * [35] V. Wan, D. MacLachlan, F. Dunne, A stored energy criterion for fatigue crack nucleation in polycrystals, International Journal of Fatigue 68 (2014) 90–102.
  * [36] Y. Guan, B. Chen, J. Zou, T. B. Britton, J. Jiang, F. P. Dunne, Crystal plasticity modelling and hr-dic measurement of slip activation and strain localization in single and oligo-crystal ni alloys under fatigue, International Journal of Plasticity 88 (2017) 70–88.
  * [37] B. Chen, J. Jiang, F. P. Dunne, Is stored energy density the primary meso-scale mechanistic driver for fatigue crack nucleation?, International Journal of Plasticity 101 (2018) 213–229.
  * [38] E. Charkaluk, A. Constantinescu, An energetic approach in thermomechanical fatigue for silicon molybdenum cast iron, Materials at high temperatures 17 (3) (2000) 373–380.
  * [39] D. Cojocaru, A. Karlsson, Assessing plastically dissipated energy as a condition for fatigue crack growth, International Journal of Fatigue 31 (7) (2009) 1154–1162.
  * [40] A. Cruzado, S. Lucarini, J. LLorca, J. Segurado, Microstructure-based fatigue life model of metallic alloys with bilinear coffin-manson behavior, International Journal of Fatigue 107 (2018) 40–48.
  * [41] A. Cruzado, S. Lucarini, J. LLorca, J. Segurado, Crystal plasticity simulation of the effect of grain size on the fatigue behavior of polycrystalline inconel 718, International Journal of Fatigue 113 (2018) 236–245.
  * [42] A. Rovinelli, R. A. Lebensohn, M. D. Sangid, Influence of microstructure variability on short crack behavior through postulated micromechanical short crack driving force metrics, Engineering Fracture Mechanics 138 (2015) 265–288.
  * [43] G. M. Castelluccio, D. L. McDowell, Mesoscale modeling of microstructurally small fatigue cracks in metallic polycrystals, Materials Science and Engineering: A 598 (2014) 34–55.
  * [44] S. Lucarini, J. Segurado, On the accuracy of spectral solvers for micromechanics based fatigue modeling, Computational Mechanics 63 (2) (2019) 365–382.
  * [45] V. Prithivirajan, M. D. Sangid, The role of defects and critical pore size analysis in the fatigue response of additively manufactured in718 via crystal plasticity, Materials and Design 150 (2018) 139–153.
  * [46] A. Eghtesad, M. Knezevic, A full-field crystal plasticity model including the effects of precipitates: Application to monotonic, load reversal, and low-cycle fatigue behavior of inconel 718, Materials Science and Engineering: A 803 (2021) 140478.
  * [47] J. Cheng, P. Fernandez-Zelaia, X. Hu, M. Kirka, Effect of microstructure on fatigue crack propagation in additive manufactured nickel-based superalloy haynes 282: an experiment and crystal plasticity study, Journal of Materials Science (2022).
  * [48] S. Ghorbanpour, J. Bicknell, M. Knezevic, Fatigue strength of additive manufactured mar-m-509 superalloy, Materials Science and Engineering: A 840 (2022) 142913.
  * [49] M. Cao, Y. Liu, F. P. Dunne, A crystal plasticity approach to understand fatigue response with respect to pores in additive manufactured aluminium alloys, International Journal of Fatigue (2022) 106917.
  * [50] T. Ozturk, A. D. Rollett, Effect of microstructure on the elasto-viscoplastic deformation of dual phase titanium structures, Computational Mechanics 61 (1-2) (2017) 55–70.
  * [51] C.-Y. Ou, R. Voothaluru, C. R. Liu, Fatigue crack initiation of metals fabricated by additive manufacturing—a crystal plasticity energy-based approach to IN718 life prediction, Crystals 10 (10) (2020) 905\.
  * [52] V. Prithivirajan, M. D. Sangid, The role of defects and critical pore size analysis in the fatigue response of additively manufactured IN718 via crystal plasticity, Materials and Design 150 (2018) 139–153.
  * [53] W. Ye, P. Kumar, L. T. Mushongera, Understanding the low cycle fatigue of additive manufactured inconel 718: a crystal plasticity modelling approach, International Journal of Theoretical and Applied Multiscale Mechanics 3 (4) (2021) 313.
  * [54] J. Zhang, J. Li, S. Wu, W. Zhang, J. Sun, G. Qian, High-cycle and very-high-cycle fatigue lifetime prediction of additively manufactured alsi10mg via crystal plasticity finite element method, International Journal of Fatigue 155 (2022) 106577.
  * [55] S. Han, X. Yang, D. Shi, G. Miao, J. Huang, R. Li, Microstructure-sensitive modeling of competing failure mode between surface and internal nucleation in high cycle fatigue, International Journal of Plasticity 126 (2020) 102622.
  * [56] C. M. Pilgar, A. Fernandez, S. Lucarini, J. Segurado, Effect of printing direction and thickness on the mechanical behaviour of SLM fabricated hastelloy-x, International Journal of Plasticity (2022) 103250.
  * [57] A. S. Karapuzha, T. Wegener, M. Krochmal, Y. Zhu, T. Niendorf, D. Fraser, X. Wu, A. Huang, Fatigue crack growth in additively manufactured hastelloy x \- influences of crack orientation and post-fabrication treatments, Materials Science and Engineering: A 854 (2022) 143773.
  * [58] F. Bachmann, R. Hielscher, H. Schaeben, Grain detection from 2d and 3d ebsd data—specification of the mtex algorithm, Ultramicroscopy 111 (12) (2011) 1720–1733.
  * [59] S. Lucarini, J. Segurado, An algorithm for stress and mixed control in galerkin-based FFT homogenization, International Journal for Numerical Methods in Engineering 119 (8) (2019) 797–805.
  * [60] M. A. Lopez-Sanchez, Grainsizetools: a python script for grain size analysis and paleopiezometry based on grain size, Journal of Open Source Software 3 (30) (2018) 863.
  * [61] A. Cruzado, J. LLorca, J. Segurado, Modeling cyclic deformation of inconel 718 superalloy by means of crystal plasticity and computational homogenization, International Journal of Solids and Structures 122-123 (2017) 148–161.
  * [62] N. Ohno, J.-D. Wang, Kinematic hardening rules with critical state of dynamic recovery, part II: Application to experiments of ratchetting behavior, International Journal of Plasticity 9 (3) (1993) 391–403.
  * [63] C. Tome, G. Canova, U. Kocks, N. Christodoulou, J. Jonas, The relation between macroscopic and microscopic strain hardening in fcc polycrystals, Acta metallurgica 32 (10) (1984) 1637–1653.
  * [64] J. Vondřejc, J. Zeman, I. Marek, An FFT-based galerkin method for homogenization of periodic media, Computers and Mathematics with Applications 68 (3) (2014) 156–173.
  * [65] J. Zeman, T. W. J. de Geus, J. Vondřejc, R. H. J. Peerlings, M. G. D. Geers, A finite element perspective on nonlinear FFT-based micromechanical simulations, International Journal for Numerical Methods in Engineering 111 (10) (2017) 903–926.
  * [66] T. W. J. de Geus, J. Vondřejc, J. Zeman, R. H. J. Peerlings, M. G. D. Geers, Finite strain FFT-based non-linear solvers made simple, Computer Methods in Applied Mechanics and Engineering 318 (2017) 412–430.
  * [67] F. Willot, B. Abdalla, Y. Pellegrini, Fourier-based schemes with modified green operator for computing the electrical response of heterogeneous media with accurate local fields, International Journal for Numerical Methods in Engineering 98 (7) (2014) 518–533.
  * [68] Y. Xu, A non-local methodology for geometrically necessary dislocations and application to crack tips, International Journal of Plasticity 140 (2021) 102970\.
  * [69] D. S. Joseph, P. Chakraborty, S. Ghosh, Wavelet transformation based multi-time scaling method for crystal plasticity FE simulations under cyclic loading, Computer Methods in Applied Mechanics and Engineering 199 (33-36) (2010) 2177–2194.
  * [70] H. A. Canistraro, E. H. Jordan, S. Shixiang, L. H. Favrow, F. A. Reed, Elastic constants of single crystal hastelloy x at elevated temperatures (1998).
  * [71] V. Herrera-Solaz, J. LLorca, E. Dogan, I. Karaman, J. Segurado, An inverse optimization strategy to determine single crystal mechanical behavior from polycrystal tests: Application to AZ31 Mg alloy, International Journal of Plasticity 57 (2014) 1 – 15.
  * [72] P. Bayati, A. Jahadakbar, M. Barati, M. Nematollahi, L. Saint-Sulpice, M. Haghshenas, S. A. Chirani, M. J. Mahtabi, M. Elahinia, Toward low and high cycle fatigue behavior of SLM-fabricated NiTi: Considering the effect of build orientation and employing a self-heating approach, International Journal of Mechanical Sciences 185 (2020) 105878.