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Article
Journal of Intelligent Material Systems
and Structures
24(17) 2097–2109
ÓThe Author(s) 2012
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DOI: 10.1177/1045389X12457835
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Crack initiation and fatigue life
prediction on aluminum lug joints
using statistical volume element–based
multiscale modeling
Jinjun Zhang, Kuang Liu, Chuntao Luo and Aditi Chattopadhyay
Abstract
This article presented the application of an energy-based multiscale damage criterion for crack initiation and life predic-
tion in crystalline metallic aerospace structural components under fatigue loading. A novel meso statistical volume ele-
ment model was developed to improve computational efficiency compared to traditional meso representative volume
element models. The key microscale factors affecting the mechanical properties of crystalline materials, including grain
orientation, misorientation, principal axis direction, size, aspect ratio, and shape were considered in the formation of the
statistical volume element model. The effect of several factors was studied to assess the importance in the overall
macroscopic response of the material. Fatigue tests of lug joint samples were performed to validate the damage criterion
as well as the statistical volume element model. Crack initiation was predicted within 29% accuracy, and orientation was
predicted within a 2°range, which was comparable to other methods. The simulation efficiency of the statistical volume
element model was improved 15 times over the traditional representative volume element models.
Keywords
Structural health monitoring, statistical volume element, multiscale modeling
Introduction
Damage detection, accurate material modeling tech-
niques, and structural health monitoring (SHM) are
emerging technologies critical to both current and
future aerospace vehicles (Bond and Farrow, 2000;
Carol and Bazant, 1997; Groh et al., 2009; Liu et al.,
2010, 2011, 2012; Mohanty et al., 2009; Yekani Fard
et al., 2012a, 2012b, 2012c). A significant amount of
research has been conducted in this field to improve the
fidelity of damage assessment models, particularly in
metallic materials (Clayton et al., 2004; Fan et al., 2001;
Hochhalter et al., 2010; Horstemeyer and McDowell,
1998; Sundararaghavan and Zabaras, 2008). However,
defects, such as cracks, initiate at the microscale before
manifesting at the macroscale and can become a critical
factor in final structural failure. Prediction of damage
initiation and crack growth can be simulated and veri-
fied by applying physics-based multiscale models. A
broad range of multiscale modeling techniques has been
reported in recent literature, and within these, the meso
representative volume element (RVE) approach is a
popular technique (Balzani et al., 2009; Kanit et al.,
2003; Trias et al., 2006). In previous research developed
by the authors (Luo and Chattopadhyay, 2011; Luo
et al., 2009), a meso RVE was directly obtained through
microstructure scans of actual material samples con-
taining various differently oriented grains. The material
behavior in hot spots, areas of high stress concentra-
tions, was described by crystal plasticity with an expli-
citly modeled microstructure in order to capture grain
size and orientation effects, while homogenous material
properties were used outside the hot spots. This multi-
scale model captures anisotropic behavior at the micro-
scale due to crystalline orientations while maintaining
the overall (homogenized) material behavior at larger
scales in accordance with the isotropic macroscale
material behavior obtained from experiments. While
this type of approach has qualified accuracy, it requires
School for Engineering of Matter, Transport, and Energy, Arizona State
University, Tempe, AZ, USA
Corresponding author:
Jinjun Zhang, School for Engineering of Matter, Transport, and Energy,
Arizona State University, Tempe, AZ 85287, USA.
Email: jzhan134@asu.edu
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large amounts of experimental data and associated pre-
processing. Also, for accuracy, the material scan has to
be acquired from the same exact location where the
RVE is positioned. In addition, such an approach is
deterministic in nature; therefore, for a different sam-
ple, the RVE model must be reconstructed based on a
new material scan. This is time-consuming, central pro-
cessing unit (CPU) intensive, and more challenging in
applications with limited accessibility.
Statistical volume element (SVE) models have been
developed to study polycrystalline microstructures. A
considerable amount of research has been reported on
SVE models in the context of stochastic modeling.
Groeber et al. (2007, 2008) made a significant improve-
ment on characterization of three-dimensional (3D)
polycrystalline microstructures and 3D SVE modeling.
Grain features, including orientation, misorientation,
and grain size were considered in their generation of the
SVE model. SVE models have also been applied in the
prediction of crack initiation and fatigue life. Yin et al.
(2008) developed an SVE method to analyze, quantify,
and calibrate microstructure–constitutive property rela-
tions using statistical means. Voids (defects) were ran-
domly generated in their SVE model prior to studying
the material’s constitutive response. However, the crack
incubation and nucleation stages were not studied in
this work. Hochhalter et al. (2010) focused on a com-
prehensive study of crack nucleation prediction and
growth in grain boundaries. Grain features, including
grain orientation, misorientation, and boundary shape
were considered in their model, and based on their
results, it was seen that grain orientation has a signifi-
cant effect on the nucleation metrics. Although the
focus of their study was on microstructurally small fati-
gue cracks (length in the order of 10 mm), their work
could be used in further construction of SVE models
and mesoscale crack prediction (up to 1-mm length).
Guilleminot et al. (2011) generated an SVE model using
strain energy density criteria. This SVE model showed a
more accurate response to mechanical behavior because
the construction of the SVE is based on mechanical
properties instead of only geometric features of the
material. The limitation of this SVE model, however,
was that it was only valid in the elastic stage. In addi-
tion to capturing statistical effects, the use of SVEs may
also reduce preprocessing and computational effort.
The methodology outlined in this article results in a sig-
nificant increase in computational efficiency through a
reasonable reduction of elements in the analysis, which
is essential for the use of multiscale models in SHM
applications.
This article presents the development of a new SVE
model and its use to predict the fatigue life of
Aluminum 2024. Information, including material prop-
erties, orientations, and grain geometric features, which
are determined from microstructural scans, is also pre-
sented. Some of the important factors that govern the
mechanical behaviors of SVEs including grain misor-
ientation, principal axis direction, size, aspect ratio,
and shape are investigated. A parametric feature study
is conducted to investigate both local and global
responses in order to identify the critical factors. The
novel characteristics of the SVE model presented in this
article are (a) significant reduction in computational
time required when compared to traditional RVE mod-
els and (b) reduced preprocessing time by eliminating
the need to obtain direct scans of the material within
the component for every simulation.
Multiscale modeling framework
A multiscale modeling framework that bridges the crys-
talline grain microstructure of an aluminum alloy (Al
2024 T351) to the structural response of a component
subject to fatigue loading is developed. The goal is to
capture damage initiation at the microscale and surmise
its effects at the structural level and on the resulting
fatigue life. At the microscale, the response within a sin-
gle grain is treated as a continuum-level material
response and is represented by single-crystal plasticity
theory combined with fatigue damage initiation criteria.
At the mesoscale, the SVE represents the microstruc-
ture of the aluminum alloy, taking into consideration
key features including grain orientation, misorientation,
principal axis direction, size, aspect ratio, and shape.
The macroscale then captures the response of the struc-
tural component that is subjected to steady fatigue
loading. In this article, a lug joint (Figure 1) is used as
the structural component at the macroscale. The mesos-
cale and macroscale are linked together in a finite ele-
ment analysis through a two-scale mesh, where the SVE
of the microstructure is explicitly modeled and meshed
at a structural hot spot within a finite element mesh of
the component. This avoids the need for homogeniza-
tion/localization schemes to connect the two length
scales. The microscale bridges to the mesoscale through
Figure 1. Boundary conditions and implementation of a two-
scale mesh at the structural hot spot of an aluminum lug joint.
SVE: statistical volume element.
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a user-defined material subroutine (UMAT) governing
the stress–strain response of an element within the SVE.
The overall multiscale modeling framework is then
applied toward the problem of efficiently predicting
crack initiation and direction within the component.
Microscale material modeling—single-crystal
plasticity and multiscale damage criterion
The authors have previously established the need for
numerical implementation of single-crystal plasticity in
multiscale modeling problems. The details of this work
can be found in their earlier articles (Luo and
Chattopadhyay, 2011; Luo et al., 2009). The most
important feature of single-crystal plasticity is its ability
to capture the local anisotropic behavior due to varying
grain orientations. In previous research, single-crystal
plasticity has been applied to microstructures obtained
directly from electron backscatter diffraction (EBSD)
scans. This article employs the use of an SVE, where
each statistically representative grain is governed by
single-crystal plasticity. In addition, the developed mul-
tiscale damage model provides damage parameters
bridging the microscale to the mesoscale.
The multiscale damage criterion presented in this
article is modified from the earlier work of the authors
(Luo and Chattopadhyay, 2011; Luo et al., 2009). Here,
the critical damage value has been improved to accom-
modate SVEs. The multiscale fatigue damage criterion,
which was modified from Jiang’s (2000) model and
extended to the microscale with single-crystal plasticity
theory (Asaro and Rice, 1977; Hill, 1966; Rice, 1971), is
presented next.
The modified energy-based damage parameter for
each slip system for a single crystal is shown in equa-
tion (1)
dD(a)=\smr
s0
1.m(1+sn(a)
sf
)dY (a)ð1Þ
where dD
(a)
represents the damage parameter incre-
ment on the ath slip system, s
mr
represents the memory
stress, s
0
represents the endurance limit, \.is the
Macaulay bracket, mis a material constant, sn(a)repre-
sents the normal stress on the ath slip system, s
f
repre-
sents the true fracture stress, and dY (a)represents the
plastic strain energy increment on the ath slip system.
Here, dY (a)can be expressed as
dY (a)=bsn(a)(dep)(a)+1b
2ssaðÞ
(dgp)(a)ð2Þ
where bis a material constant, deprepresents the plastic
strain increment corresponding to sn(a),ss(a)represents
the shear stress on slip system a, and dg
p
represents the
plastic strain increment corresponding to ss(a). Face-
centered cubic (FCC) crystal structures, such as
aluminum, are of interest in this article; therefore, 12
slip systems are considered here. Using equations (1)
and (2), the damage parameter dD(a)for each slip sys-
tem, a, is calculated. In order to fuse the damage infor-
mation of all 12 slip systems, a damage tensor is
developed to indicate the damage state at the conti-
nuum level. In equation (1), the damage parameters in
the 12 slip systems are in a strain energy density form.
Based on this, a symmetric damage tensor is assumed in
order to reflect the directional effects for different slip
systems. Thus, the overall damage tensor increment
within a single crystal is related to the slip system dam-
age parameter increment as follows
dD(a)=h(a)dDs(a),a=1to 12 ð3Þ
where h(a)is the unit vector in the normal direction of
the ath slip system, dDis the damage tensor increment,
and s(a)is the unit vector in the slip direction of the ath
slip system. For the symmetric damage tensor, there
are six independent components. However, the 12 con-
ditions resulting from equation (3) overconstrain the
problem. In order to find a unique tensor that satisfies
all conditions simultaneously, the damage tensor is
computed using an optimization technique. The objec-
tive function in incremental form is expressed as
E2=X
12
a=1
½dD aðÞ
h(a)dDs(a)2ð4Þ
The six components of the damage tensor for each
integrated point in SVE model are defined by solving
for the minimum of the objective function.
In order to obtain the magnitude and direction of
damage at the mesoscale, first the information on dam-
age levels within a grain is needed. However, because
of stress distributions within the grain, the damage is
not constant. To circumvent this, the damage tensors
computed at each integration point within the grain are
averaged over the entire grain, that is
Dg=P
N
n=1
D
Nð5Þ
where the overbar denotes the average quantity and N
represents the number of integration points within a
grain. Next, the average deviatoric damage tensor for
each grain is calculated as
dgij =
Dgij 1
3dij
Dgkk ð6Þ
where
dgij denotes the component of average deviatoric
damage tensor for each grain,
Dgij denotes the compo-
nent of average damage tensor for each grain, i= 1–3
and j= 1–3. For each average deviatoric damage ten-
sor, there is a maximum eigenvalue l1and a
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corresponding eigenvector ~
v1, which are used to define
the average damage vector within a grain, D
*
g. The
average damage vector for a grain is defined as
~
Dg=l1~
v1
~
v1
kk ð7Þ
Here, the magnitude of the damage vector D
*
gk
for
each grain is set equal to the maximum eigenvalue of
the average deviatoric damage tensor. The damage vec-
tor lies along the corresponding eigenvector of the max-
imum eigenvalue. The upper and lower bounds for
D
*
g
of all grains in the meso SVE area are denoted as
Duand Dl, respectively. The increment in damage at
the mesoscale can be calculated by
dDmeso =udDu+(1u)dDlð8Þ
where uis a weight factor that measures the area of
grains in which D
*
g
reaches the critical damage value
Dc. The weight factor uis defined as
u=a
Að9Þ
where ais the total area of grains that have reached the
critical damage value and Ais the entire area of the
SVE. For the SVE model, each grain has the same area.
Therefore, the weight factor is modified as follows
u=n
Nð10Þ
where nis the number of grains that have reached the
critical damage value and Nis the total number of
grains in the SVE. The critical damage value for each
grain and for the complete SVE is determined by
Dc=23g3
l
Að11Þ
where D
c
represents the critical damage value, grepre-
sents the surface energy density, lrepresents the average
grain size, and Arepresents the meso SVE area. When
a grain reaches the critical damage value D
c
,nincreases
by one and the weight factor uincreases by 1/N, which
in turn increases D
meso
. The SVE is considered to be
totally damaged when D
meso
reaches D
c
. The material
parameters used in equations (1) to (11) are listed in
Table 1 (Luo et al., 2009).
Mesoscale/macroscale—finite element
implementation
The mesoscale and macroscale analyses are conducted
using finite element analysis as implemented in the com-
mercial finite element solver Abaqus (Version 6.10.1).
The two length scales are solved concurrently using a
two-scale mesh, that is, a relatively small element for
the mesoscale SVE and a larger element for the remain-
der of the component. The SVE is only implemented
within the structural hot spot, a location with the high-
est damage and probability of failure, to increase com-
putational efficiency. Within the SVE, all the elements
are modeled using the previously described single-crystal
plasticity theory and damage initiation criteria. Outside
the SVE, in the far field stress regions, the elements are
governed by conventional J
2
plasticity theory based on
the homogeneous response of the aluminum alloy. The
physical size of the SVE that is implemented at the
macroscale was determined by the minimum size neces-
sary for representing orientation information of this
material. This resulted in a 1 31-mm SVE containing
approximately 40 grains. Details on the statistical for-
mulation of the SVE are presented in the next section.
The hot spot in the lug joint was determined through a
stress analysis to find the location with the highest stress
concentration. Figure 1 shows the location of the SVE
and the loading boundary conditions of the sample. The
dimensions of the lug joint sample are shown in Figure
2. The loading condition is a cyclic tensile load ranging
from 489 to 4890 N with a frequency of 20 Hz.
Prediction of fatigue life using SVE
The procedure to obtain the damage tensor described
previously was implemented into a UMAT (Huang,
1991), and a data processing technique was also devel-
oped in MATLAB. Based on the finite element analysis
results, the damage evolutions over 10 cycles in the
SVE are shown in Figure 3 (only 10 of the total grains
are shown for clarity). It can be observed from this fig-
ure that the damage evolution in each grain is approxi-
mately linear from cycle to cycle. This was validated
with simulations up to 30 cycles. This observation pro-
vides the basis for life prediction. Using the results from
the first 10 cycles, the rate of damage evolution of all
the grains in the SVE is calculated. When the magni-
tude of damage tensor in each grain reaches D
c
, the
weight factor uwill increase by 1/Nand thus D
meso
will
increase as well. When D
meso
reaches D
c
, the SVE is
considered ‘‘failed’’ and the crack is considered to pro-
pagate through the SVE model. The length of the crack
is considered to be the same as the SVE dimension, that
is, 1 mm. Although the damage may grow nonlinearly
after 10 cycles, a linear growth assumption is utilized to
calculate the damage growth efficiently.
Table 1. Material parameters for Al 2024 T351 (Luo et al.,
2009).
mbs
0
s
f
g
1.5 0.32 220 MPa 950 MPa 865.18 MJ/m
2
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The direction of damage tensor D
*varies with time
during the simulation. The distribution of damage
directions versus loading time is shown in Figure 4.
From the distribution results, the most probable dam-
age direction is considered as the potential crack direc-
tion. More details about this damage criterion can be
Figure 2. Dimensional drawing of lug joint and rolling direction of material processing. The rolling direction affects grain features.
Figure 4. Potential crack direction over time. Clustering of
spots around any degree represents a probable crack direction.
Figure 3. Linear growth in grain-level damage parameter.
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found in authors’ earlier work (Luo and Chattopadhyay,
2011; Luo et al., 2009).
SVE formulation
Construction of SVE
In this section, the methodology used to construct the
SVE used for the multiscale analysis and fatigue life
prediction is outlined. The primary goal of this research
is to improve the computational efficiency and reduce
the preprocessing effort associated while maintaining
accuracy. This is achieved by employing simplified grain
shapes, which result in ease of assembly, reduction in
preprocessing time, and a reduction in total number of
elements (irregular grain shapes result in small features
which require small elements). The SVE is constructed
by assembling grains whose features are statistically
sampled from pools of measured experimental charac-
terization data (Figure 5(b)). This approach provides a
computationally efficient alternative to traditional tech-
niques while maintaining statistical accuracy.
The nonstatistical parameters, including grain size,
shape, aspect ratio, and principal axis direction, were
carefully chosen to be representative of the average
response of the material. The grain size was chosen to
be 0.025 mm
2
, while the measured average value was
0.0225 mm
2
. This is to ensure that an exact number of
grains fit within a 1 31-mm SVE. A rectangular grain
shape is used to efficiently assemble the grains within
the square SVE. The aspect ratio was chosen to be 0.4,
while the average measured value was 0.384, for the
same reason as previously stated. Finally, the principal
axis direction was chosen to align with the rolling direc-
tion as shown in Figure 2. The final architecture of the
SVE can be seen in Figure 5(b). As mentioned previ-
ously, the orientation was chosen based on the statisti-
cal features and as such its distribution was studied.
Figure 6 shows three pole figures for grain orienta-
tions. The distributions do not exhibit any trends and
therefore are assumed to be normally distributed. It
must be noted that when formulating the SVE, the
orientations were chosen at random from the pool of
available material characterization data. The last fea-
ture, misorientation, is determined from the arrange-
ment of the grains in the SVE. Since the ordering can
be controlled, the misorientation was chosen to closely
match that of the experimental data shown in Figure
7(a). To accomplish this, nine randomly orientated
SVEs were generated. A typical histogram of their mis-
orientation is shown in Figure 7(b). The difference, that
is, error between the randomly generated SVEs and the
experimental data, is shown in Figure 8. The seventh
generated SVE is the closest match to the experimental
data and therefore was used in the analysis.
Feature parametric study
To understand the effects and sensitivity of each feature
used in the SVE formulation, several parametric studies
were conducted. The features considered in this study
were grain orientation, misorientation (Groeber et al.,
2007; Wang et al., 2009), principal axis direction, size,
aspect ratio, and shape. In order to create a true SVE,
each feature has to be represented by an appropriate
statistical observation. However, this process can be
complex and time-consuming. In this work, only grain
orientation was chosen as a statistical feature because it
is a primary factor in single-crystal plasticity that results
in damage initiation (Asaro and Rice, 1977; Hill, 1966;
Figure 5. Comparison between (a) actual microstructure scan and (b) constructed SVE. Colors represent different grain
orientations.
SVE: statistical volume element.
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Hochhalter et al., 2010). Experimental observation
has shown that these data are randomly distributed.
The remaining features, except misorientation, are
geometric features and can be implemented as a
constant value based on the average experimental
observation. To validate this assumption, a feature
study was performed to evaluate the effect of each
geometric feature.
Figure 6. Flattened 3D polar plots of crystal orientation: (a) {1,0,0}, (b) {1,1,0}, and (c) {1,1,1}.
3D: three-dimensional.
Figure 7. Statistical distribution used for misorientation feature identification: (a) actual data and (b) example of generated SVE.
SVE: statistical volume element.
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The feature data were obtained for a total of 513
grains using EBSD scans. To ascertain the effect of
each feature, one baseline SVE and five other SVEs,
each emphasizing a different feature, were constructed.
The baseline SVE (SVE-A), shown in Figure 9, was
constructed using a representative value for each geo-
metric feature. These values were obtained via random
sampling and ordering of grain orientation, resulting in
an approximate random misorientation. A change in a
single variable from the reference SVE leads to a new
SVE. For example, SVE-B was created to investigate
the effect of misorientation; therefore, the order of the
grains was changed, but their size, shape, aspect ratio,
and principal axis direction were held constant.
Similarly, the role of SVE-C is to investigate the effect
of a change in principal axis direction, while its misor-
ientation is maintained as close as possible to SVE-A.
SVE-D considers grain size by increasing the number
of grains fourfold while keeping all other factors con-
stant. SVE-E helps to analyze the aspect ratio by
increasing the number of columns and reducing the
number of rows resulting in a 1:4 aspect ratio. The role
of SVE-F is to explore grain shape by modeling a hexa-
gon instead of a square. Figure 9 shows the SVE
Figure 8. Comparison of the SVE models with reference SVE: (a) difference versus orientation and (b) average errors.
SVE: statistical volume element; RMS: root mean square.
Figure 9. SVEs constructed for feature study: (a) baseline, (b) misorientation, (c) principal axis direction, (d) grain size, (e) aspect
ratio, (f) grain shape. Feature effects could be studied through comparison of results with respect to the baseline.
SVE: statistical volume element.
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configurations with the numbers indicating particular
grain orientations. The grain size shown is not repre-
sentative of those used in the analysis but instead
demonstrates its relative effects. Approximately, 100
representative grains of the same size were used,
except in SVE-D. The size of the SVE was held con-
stant at 1 31mm
2
.
Each SVE was analyzed by performing a multiscale
analysis. A cyclic tensile loading (489–4890 N) was
used and the local plastic strain energy densities within
the SVEs were compared. Figure 10 shows the average
plastic strain energy density of all six SVEs with respect
to time for the first loading cycle. From equations (1)
and (2), the damage is influenced greatly by accumula-
tion of plastic strain energy. It is observed that the plas-
tic strain energy increases with loading time and
reaches the maximum value at the end of the loading
history; therefore, the SVEs were compared at this
point. The difference between SVE-A and SVE-C was
2.90% and between SVE-A and SVE-E was 0.01%,
shown in Table 2. Figure 11 shows the local plastic
strain energy density fields within each SVE. The distri-
butions of plastic strain energy density in all the SVEs
are similar. Additionally, for different SVEs, the maxi-
mum local plastic strain energy densities appear at the
right edge of the SVE. Some interesting observations
can be made from these results. The highest average
density appears in SVE-C, but the highest local maxi-
mum density appears in SVE-F. The lowest average
density appears in SVE-A, but the lowest local maxi-
mum density appears in SVE-B. This shows that the
plastic strain energy density is variable for different
points in SVE model and it is a local value. Based on
the performed simulations, this implies that the effect
of these factors may be a local phenomenon, not a glo-
bal one. It must be noted that the damage initiation cri-
teria depend on the response of the SVE as a whole
and not on the response of a single grain; therefore, the
overall effect, considering all grains, must be consid-
ered in life prediction. Based on the result, the average
difference in plastic strain energy densities is small
when compared to the variability in experimental data,
and therefore, each feature can be effectively repre-
sented by its average value.
Validation of SVE model
To validate the SVE-based multiscale modeling
approach, two measures were considered: hot spot
location and far field distributions. To ensure that the
SVE is representative of the bulk material and mechan-
ical responses, both measures must remain consistent
with or without an SVE implemented in the finite ele-
ment model. The SVE is expected to have a direct influ-
ence on the local fields near the hot spot due to local
crystal plasticity effects. Without this, there would be
no means to calculate the damage at the microscale for
fatigue life estimation. Figure 12 shows comparison of
von Mises stress distributions in the lug joint models,
with and without an SVE. The overall contours show
similar trends, that is, the implemented SVE has a
small effect on the mechanical response. The hot spot
location remains unchanged at the inside corner radius
below the loading pins, and the magnitude remains
approximately the same. The distribution of stress in
the far field outside the hot spot position also remains
the same. With both the hot spot and far field values
nearly unchanged, the SVE-based multiscale modeling
technique is proven capable of representing the homo-
geneous material.
Simulation and experimental results
Experimental test and results
A series of experiments were conducted to validate the
fatigue life prediction capabilities of the statistical mul-
tiscale model. Lug joint specimens were machined from
a bulk Al 2024 T351 plate with the rolling direction
lengthwise along the lug joint. The specimens were
polished in order to clearly capture crack initiation and
Table 2. Summarized SVE feature values.
Features Misorientation Principal axis direction Grain size Aspect ratio Grain shape
Average percent difference 1.70 2.90 0.30 0.01 1.80
SVE: statistical volume element.
Figure 10. Variation in average plastic strain energy density
versus time.
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growth. The specimens were cyclically loaded in a servo
hydraulic desktop test frame using a sinusoidal wave-
form between 489 and 4890 N (load ratio 0.1) at a rate
of 20 Hz.
To track the crack growth, a camera was mounted
and images were captured every 1000 cycles. The crack
length was computed through digital measurements
using calibrated images. To validate the multiscale
model, the number of cycles necessary to achieve a
1-mm crack was obtained experimentally. Using this
experimental approach, the estimated number of cycles
was found to be accurate to within 1000 cycles. The
crack initiation angle was calculated by averaging the
angle formed along the crack path from the initial (0
mm) to final (1 mm) points. A total of five specimens
were tested. The experimental results are summarized
in Table 3. The final image at failure of each specimen
illustrating the crack angle is shown in Figure 13. The
mean deviation for the number of cycles to produce a
1-mm crack was 84.6 k cycles, and the average crack
direction was 228.8°; the coordinate system is shown
in Figure 13.
Figure 11. Plastic strain energy density distributions within SVEs: (a) baseline, (b) misorientation, (c) principal axis direction, (d)
grain size, (e) aspect ratio, and (f) grain shape.
SVE: statistical volume element.
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Simulation results
The simulations were conducted using the same proper-
ties and conditions as the experiment. Figure 14 shows
the growth of the damage index as a function of cycles.
When the damage index reaches a value of 1, this corre-
sponds to a crack of 1-mm length. It is important to
Figure 12. Local von Mises stress field distribution within lug joint (a) with SVE and (b) without SVE.
SVE: statistical volume element.
Table 3. Experimental results of fatigue life and crack direction.
Sample 1 2 3 4 5 Average
Fatigue life (number of cycles) (k cycles) 92 98 68 83 82 84.6
Crack direction (°)225 234 229 228 228 228.8
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note that the damage index is a measure of the damage
growth within the microstructure and not a measure of
crack length; therefore, its value cannot be directly cor-
related to any other crack length besides 1 mm. The
multiscale model predicts a damage index of 1 (i.e. a 1-
mm crack) at 108.5 k cycles. Figure 15 shows the histo-
gram of simulation results with the possible crack direc-
tions. There were two probable crack direction ranges
found: 235°to 230°and 70°to 75°based on the coor-
dinate system shown in Figure 13. The 235°to 230°
range was chosen as the most probable because it was
most closely aligned with the direction perpendicular to
maximum principal stress. Compared with the experi-
mental result, there is a 28.3% error in the fatigue life
prediction. All crack directions are in or close to one of
the direction ranges of the simulation result (235°to
230°). The error in fatigue life prediction can be attrib-
uted to several possible reasons. First, the accuracy of
the surface energy density of a bulk homogenized mate-
rial is most likely not the same as that within a grain.
Also, the result of the life prediction, which is 108.5 k
cycles for a 1-mm crack, represents an ideal material
status. The presence of defects or flaws in the real mate-
rial will shorten the fatigue life. However, by simplify-
ing the SVE based on fixed grain sizes and regular
shapes, a significant reduction of about 87.40% was
achieved in the total number of the elements used, from
14,606 (Luo et al., 2009) to 1840. The previously devel-
oped RVE model (Luo et al., 2009) had errors in the
same order of magnitude, but required computation
time that was more than 1 order of magnitude greater.
Conclusion
A multiscale modeling framework based on a SVE was
developed to predict the fatigue life and crack direction
within an aluminum component. Three length scales, the
microscale, mesoscale, and macroscale, were considered
in this analysis. At the microscale, the material response
within a single crystal was modeled using the modified
single-crystal plasticity theory. At the mesoscale, the
material microstructure was captured by implementing a
computationally efficient SVE model. The SVE captured
the statistical distribution of orientation and misorienta-
tion while implementing the average response of geome-
trical features, including grain principal axis direction,
size, shape, and aspect ratio. A parametric study was per-
formed to assess the sensitivity of each factor. Principal
axis direction and misorientation were determined to be
two of the most important features affecting the mechani-
cal response of the constructed SVE. At the macroscale, a
finite element analysis of a lug joint subjected to cyclical
tensile loads was conducted with the mesoscale SVE
implemented through a two-scale mesh. A multiscale
damage criterion that links damage initiation at the
microscale to damage effect at the mesoscale (grain level)
was used. The developed model was capable of capturing
the crack initiation direction as well as estimating fatigue
life. The error of fatigue life prediction with SVE model is
in the same order of magnitude as previously developed
RVE model. The SVE model resulted in a significant
improvement in preprocessing efficiency as well as com-
putational efficiency through reduced number of elements
when compared to a conventional RVE-based model.
Funding
This research is supported by the Department of Defense,
AFOSR Multidisciplinary University Research Initiation
Figure 13. Crack formation in experimental samples.
Figure 14. Mesoscale damage index versus fatigue load.
Figure 15. Potential crack direction histogram. The vertical
axis shows the frequency of predicted crack directions in 10
cycles. There are two probable crack direction bands: 235°to
230°and 70°to 75°.
2108 Journal of Intelligent Material Systems and Structures 24(17)
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(MURI) Grant, FA95550-06-1-0309, Technical Monitor: Dr
David S. Stargel.
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