metals
Article
Finite Element Based Physical Chemical Modeling of
Corrosion in Magnesium Alloys
Venkatesh Vijayaraghavan 1, *, Akhil Garg 2 , Liang Gao 3 and Rangarajan Vijayaraghavan 4
1
2
3
4
*
School of Engineering, Monash University Malaysia, 47500 Bandar Sunway, Selangor Darul Ehsan, Malaysia
Department of Mechatronics Engineering, Shantou University, Shantou 515063, China; [email protected]
The State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of
Science and Technology, Wuhan 430074, China; [email protected]
Materials Engineering Department, KOP Surface Products Singapore Pte Ltd, 77 Science Park Drive 118256,
Singapore; [email protected]
Correspondence: [email protected]; Tel.: +60-355-159-711
Academic Editor: Hugo F. Lopez
Received: 18 January 2017; Accepted: 28 February 2017; Published: 7 March 2017
Abstract: Magnesium alloys have found widespread applications in diverse fields such as aerospace,
automotive, bio-medical and electronics industries due to its relatively high strength-to-weight ratio.
However, stress corrosion cracking of these alloys severely restricts their applications in several
novel technologies. Hence, it will be useful to identify the corrosion mechanics of magnesium
alloys under external stresses as it can provide further insights on design of these alloys for critical
applications. In the present study, the corrosion mechanics of a commonly used magnesium alloy,
AZ31, is studied using finite element simulation with a modified constitutive material damage model.
The data obtained from the finite element modeling were further used to formulate a mathematical
model using computational intelligence algorithm. Sensitivity and parametric analysis of the derived
model further corroborated the mechanical response of the alloy in line with the corrosion physics.
The proposed approach is anticipated to be useful for materials engineers for optimizing the design
criteria for magnesium alloys catered for high temperature applications.
Keywords: finite element analysis; corrosion mechanics; AZ31 alloy; computational intelligence
1. Introduction
Research in magnesium (Mg) alloys has found widespread interest in recent years due to their
attractive strength-to-weight ratio, lightweight and high stiffness. These extraordinary properties
make Mg alloys particularly useful for applications requiring high performance and energy savings
such as structural engineering, automotive and biomedical applications [1–3]. It should also be noted
that the weak corrosion resistance of Mg alloys restricts their applications in several novel fields [4–7].
Most of the existing studies in Mg alloys are focused on investigating the mechanical strength based
on specific applications such as structural, mechanical or bio-medical engineering. Hence, there is a
need to study the mechanics of Mg alloys under corrosive conditions. It is also well known that Mg
alloys are more susceptible to stress corrosion cracking (SCC). Hence, an effective deployment of Mg
alloys in new and emerging fields will require an understanding of the behavior of corroded Mg alloys
under mechanical and thermal loading conditions.
Most of the literature studies on Mg alloys are based on their mechanical properties for specific
applications. Argade et al. [8] studied the mechanism of AZ31 Mg alloy under slow strain rate testing
technique. They found that the corrosion of AZ31 is predominantly influenced by its microstructure
which results in a favorable texture for increased adsorption of hydrogen. Su et al. [9] deployed a
controlled phosphate deposition technique for improving the surface biocompatibility of AZ60 Mg
Metals 2017, 7, 83; doi:10.3390/met7030083
www.mdpi.com/journal/metals
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alloy. They found that the coating process is influenced by temperature and pH, which affects the
corrosion resistance of AZ60 alloy. Choudary et al. [10] investigated the SCC mechanism of various
aluminum free Mg alloys in a simulated human fluid environment. It was found that the SCC of
Mg alloys in the simulated condition were attributed to their electrochemical and microstructural
characteristics. Gaur et al. [11] developed a silane based biodegradable coating for improving corrosion
resistance of Mg alloys in physiological environment. They found that formation of a stable and
uniform hydroxide layer on alloys surface facilitated adhesion of silane coating. Wei et al. [12]
investigated the corrosion behavior of plasma formed coatings on Mg alloys with respect to the
aluminum composition in the alloy mix. It was found that the difference in porosity level of coatings is
caused by different discharging behaviors in Mg alloys.
In addition to experimental studies, some simulation studies have also been conducted to
investigate the corrosion mechanics of Mg alloys. Grogan et al. [13] developed a physical corrosion
model based on finite element (FE) technique for predicting mechanical strength of corroded metal
stents. The developed model was able to capture the change in surface due to corroding environment.
Duddu et al. [14] developed an extended FE model for modeling galvanic corrosion in magnesium
alloys. It was demonstrated that the proposed approach can be used to model crevice geometry
without remeshing. Shang et al. [15] deployed molecular dynamic simulations to study the adsorption
behavior of composite coatings on Mg alloys. The presented approach provided an effective application
based surface modification of Mg alloys. In addition to FE modeling, analytical models based
on computational intelligence techniques have also been widely used for predicting corrosion.
For instance, Narimani et al. [16] deployed artificial neural network for predicting the effects of
chemical composition and corrosion characteristics in magnesium alloy pipelines. Similarly, Zadeh
Shirazi and Mohammadi [17] used a hybrid neural network model for predicting corrosion rates in
underwater pipelines.
It is clear from the literature studies that experiments and simulation have been deployed to study
corrosive characteristics in Mg alloys, with simulation being more preferred mode of investigation.
This is because simulation techniques provide a viable alternative in saving cost related to time and
materials. The current state-of-the-art in scientific modeling of corrosion is directed at two fronts,
viz. mechanical and chemical model (Figure 1). These models are capable of generating accurate
solutions in predicting corrosion characteristics with minimal cost and high rapidity. In addition to
these science based models, process modeling based on computational intelligence technique has
also been conducted, as can be seen from the previous literature studies. The advantages of process
modeling include formation of an explicit mathematical relationship which can help in understanding
hidden relationship between various input parameters in corrosion process. It can be seen that the
science based models and process models have their own unique advantages. As a result, it will be
useful to integrate the models which can combine the advantages of accuracy of FE models with model
formation ability of process modeling.
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Figure1.
1. State-of-art
State-of-art research
mechanics
of magnesium
alloys.
Figure
researchinincorrosion
corrosion
mechanics
of magnesium
alloys.
Hence, the main objective of the present research proposes an integrated science based model
themodel
main objective
of thecorrosion
present research
proposes
an integrated
science
based
andHence,
process
for predicting
characteristics
of Mg
alloy (Figure
2). The
Mgmodel
alloy and
process
model
for
predicting
corrosion
characteristics
of
Mg
alloy
(Figure
2).
The
Mg
alloy
considered
considered in this study is the AZ31 alloy used in aircraft components, biodegradable metallic
in implants,
this studycell
is the
AZ31
alloy
used
in aircraft
biodegradable
implants,
cell phone
phone
and
laptop
cases,
etc. In components,
the present study,
a corrodedmetallic
AZ31 alloy
with cracks
and
laptop
etc.resulting
In the present
study, a corroded
AZ31
alloy(e.g.,
withhydrogen
cracks and
voids typically
and
voids cases,
typically
from an environment
that leads
to SCC
embrittlement)
is modeled
using
FE simulation.that
Theleads
corrosive
film(e.g.,
factor
is included
in the study byisconsidering
the FE
resulting
from
an environment
to SCC
hydrogen
embrittlement)
modeled using
Pilling–Bedworth
ratio
(P–B
ratio),
which
is
defined
as
the
ratio
of
the
volume
of
corrosive
metal
simulation. The corrosive film factor is included in the study by considering the Pilling–Bedworth
oxide
to the
volume
of is
thedefined
metal. As
can ratio
be seen
2, the
focuses
on oxide
understanding
the of
ratio
(P–B
ratio),
which
as the
of in
theFigure
volume
of study
corrosive
metal
to the volume
of AZ31
alloy with
respect
the input
viz. the
ratio, strain
rate of
themechanical
metal. Asstrength
can be seen
in Figure
2, the
studytofocuses
on parameters
understanding
theP–B
mechanical
strength
and alloy
temperature.
As can
seen
in Figure
2, the FE
is used
tostrain
predict
theand
mechanical
strength
AZ31
with respect
tobe
the
input
parameters
viz.model
the P–B
ratio,
rate
temperature.
As can
of AZ31 alloy in corrosive environment with respect to the input parameters, viz. the P–B ratio,
be seen in Figure 2, the FE model is used to predict the mechanical strength of AZ31 alloy in corrosive
strain rate and temperature. The process modeling based on genetic programming (GP) is the used
environment with respect to the input parameters, viz. the P–B ratio, strain rate and temperature.
to obtain the mathematical relationship between the mechanical strength based on the input
The process modeling based on genetic programming (GP) is the used to obtain the mathematical
parameters. Scientific analysis of the formulated model is also undertaken based on actual corrosion
relationship
between the mechanical strength based on the input parameters. Scientific analysis of
science which will provide vital design inputs for maximizing the performance of AZ31 alloys in
thecorrosive
formulated
model is also
undertaken
based on actual
corrosion
science whichthe
willcorrosion
provide vital
environment.
The
various approaches
deployed
in investigating
design
inputs
for
maximizing
the
performance
of
AZ31
alloys
in
corrosive
environment.
characteristics in AZ31 alloys and the proposed research objective is depicted in Figures 1The
andvarious
2,
approaches
deployed
in
investigating
the
corrosion
characteristics
in
AZ31
alloys
and
the
proposed
respectively.
research objective is depicted in Figures 1 and 2, respectively.
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Finite Element Modeling of AZ31 Alloy
23
0.8
0.7
0.6
0.5
0.4
.2
5
4
3
2
1
True stress
Input data (P-B ratio,
Temperature and Strain
rate)
Genetic Programming
Settings of GP such as population size,
generations, depth of gene, etc.
True stress
model
+
+
5
Strain rate
P-B ratio
Temperature
Figure
2. FEM-GP
approach
in modeling
corrosion
mechanics
of AZ31 alloy.
FEM-GP:
Element
Figure
2. FEM-GP
approach
in modeling
corrosion
mechanics
of AZ31
alloy. Finite
FEM-GP:
Finite
Modeling-Genetic
Programming;
P–B ratio: Pilling–Bedworth
ratio.
Element Modeling-Genetic
Programming;
P–B ratio: Pilling–Bedworth
ratio.
2. Finite
Element
Modeling
Corroded
AZ31
Alloy
2. Finite
Element
Modeling
of of
Corroded
AZ31
Alloy
2.1.2.1.
Geometry
Description
andand
Boundary
Conditions
Geometry
Description
Boundary
Conditions
The
FEFE
modeling
of AZ31
alloy
is performed
using
ABAQUS/Explicit
Version
6.14
software
The
modeling
of AZ31
alloy
is performed
using
ABAQUS/Explicit
Version
6.14
software
(Dassault
Systemes,
Vélizy-Villacoublay,
France,
2014)
with
fully
coupled
thermal
stress
analysis.
(Dassault
Systemes,
Vélizy-Villacoublay,
France,
2014)
with
fully
coupled
thermal
stress
analysis.
The
thermal
analysis
enables
modeling
thethe
strength
characteristics
of of
AZ31
alloy
at at
elevated
The
thermal
analysis
enables
modeling
strength
characteristics
AZ31
alloy
elevated
temperatures.
The
model
geometry
constructed
in
ABAQUS
consists
of
AZ31
alloy
of
dimensions
temperatures. The model geometry constructed in ABAQUS consists of AZ31 alloy of dimensions
450450
mmmm
× 20× mm
× 10 ×mm.
byproductbyproduct
of corrosion
Mg alloys in
is magnesium
20 mm
10 The
mm.most
Thepredominant
most predominant
ofincorrosion
Mg alloys is
oxide
(MgO). This
corrosive
is included
in the
adding
a film by
of MgO
on atop
of of
magnesium
oxide
(MgO).product
This corrosive
product
is geometry
included by
in the
geometry
adding
film
AZ31
alloy.
In
the
ABAQUS/CAE
model,
one
end
of
the
model
geometry
is
fixed
with
an
ENCASTRE
MgO on top of AZ31 alloy. In the ABAQUS/CAE model, one end of the model geometry is fixed
boundary
thatboundary
arrests allcondition
rotationalthat
and arrests
translational
degrees and
of freedom.
Translational
with ancondition
ENCASTRE
all rotational
translational
degrees of
degree
of
freedom
along
one
axis
is
provided
to
the
other
end
of
the
model
geometry
for
tensile
freedom. Translational degree of freedom along one axis is provided to the other end of the
model
geometry for tensile loading condition. The description of AZ31 model geometry with corrosive film
along with the applied boundary condition is depicted in Figure 3. The material properties of AZ31
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loading condition. The description of AZ31 model geometry with corrosive film along with the applied
boundaryMetals
condition
is depicted in Figure 3. The material properties of AZ31 alloy and
the corrosive
2017, 7, 83
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oxide film used in the FE modeling is illustrated in Table 1. The defined model geometry is then
alloydiscrete
and the corrosive
oxide(CPE8RT)
film used intype
the FE
modeling following
is illustratedwhich
in Table
1. The defined
meshed into
eight-node
elements,
boundary
conditions are
model geometry is then meshed into discrete eight-node (CPE8RT) type elements, following which
applied to
create
mechanical
loading
of
the
AZ31
part.
boundary conditions are applied to create mechanical loading of the AZ31 part.
Figure 3. Model geometry of AZ31 alloy and corrosive film considered in study.
Figure 3. Model geometry of AZ31 alloy and corrosive film considered in study.
Table 1. Properties of alloy and film used in FEM simulation [18,19]. FEM: Finite Element Modelling.
Table 1. Properties of alloy and
film used in FEM simulation
[18,19]. MgO
FEM:film
Finite Element Modelling.
Parameter
AZ31 Alloy
Thermal Conductivity (k)
96 W/m-K
Parameter
AZ31
Density (ρ)
1770Alloy
kg/m3
Young’s
modulus
GPa
Thermal
Conductivity
(k) (E)
96 44.8
W/m-K
Density
(ρ) ratio (υ)
1770 kg/m
Poisson’s
0.35 3
Young’sSpecific
modulus
(E)(Cp)
44.8
heat
1000GPa
J/kg/K
Poisson’s ratio (υ)
0.35
Specificofheat
(Cin
1000 J/kg/K
p ) AZ31 under Corrosion
2.2. Constitutive Modeling
Voids
42 W/m-K
3580 MgO
kg/m3Film
249 42
GPa
W/m-K
3580 kg/m3
0.18
249 GPa
877 J/kg/K
0.18
877 J/kg/K
AZ31, similar to most magnesium alloys, is susceptible to SCC. SCC results in the formation of
2.2. Constitutive
Modeling
of Voids
AZ31ofunder
Corrosion
mild cracks
and voids
on the in
surface
the material.
These cracks can nucleate and grow further
under application of external loads which results in catastrophic failure of the material. This
AZ31,
similar toinmost
is susceptible
SCC.
results
the formation
of
phenomenon
whichmagnesium
the nucleation alloys,
and growth
of cracks and to
voids
thatSCC
lead to
ductilein
failure
of
mild cracks
and
voids
on
the
surface
of
the
material.
These
cracks
can
nucleate
and
grow
further
under
metal is modeled in FE simulation using the Gurson–Tvergaard–Needleman (GTN) model [20].
Hence,
in this work,
thewhich
AZ31 alloy
subjected
to mechanical
loading
is modeled
using
the phenomenon
GTN
application
of external
loads
results
in catastrophic
failure
of the
material.
This
in
damage model with integrated Yld2000 yield criterion [21,22]. The model is programmed in
which the nucleation and growth of cracks and voids that lead to ductile failure of metal is modeled in
ABAQUS solver using the user-defined subroutine module VUMAT. This approach can provide a
FE simulation
using theofGurson–Tvergaard–Needleman
(GTN)
modelenvironment.
[20]. Hence,
this work, the
better prediction
crack evolution and failure of AZ31 alloys
in corrosive
Theinyield
AZ31 alloy
subjected
todamage
mechanical
loading
is mathematically
modeled using
the GTN damage model with integrated
function
of GTN
model is
described
as [23],
2 model is programmed in ABAQUS solver using the user-defined
Yld2000 yield criterion [21,22]. The
𝑞
−3𝑞2 𝑝
= 2 approach
+ 2𝑞1 𝑓 ∗ cosℎcan
( provide
) − (1a +better
𝑞3 𝑓 ∗2 )prediction
=0
(1)
subroutine module VUMAT.ΦThis
of crack evolution
and
2σ𝑦
σ𝑦
failure of where
AZ31σalloys
in
corrosive
environment.
The
yield
function
of
GTN
damage
model
is
described
y is the equivalent stress; p is hydrostatic stress; q is the effective stress; and q1, q2, and q3 are
mathematically
as [23], to improve prediction of void interaction effects in the Gurson damage equation.
fitting parameters
The reduction in mechanical strength of the
is defined by the
materialdue to void coalescence
∗
∗
3q2 volume
p
q2
damage parameter 𝑓 , expressed
as a function
of−
void
fraction f. ∗For
∗
2 𝑓 = 0, the GTN yield
Φ= 2 +
2q1 f cos h
− 1 +that
q3 f the material
= 0 has failed. It is
condition becomes the normal
𝑓 ∗ y= 1 indicates
σy yield condition and 2σ
defined as,
(1)
𝑓 stress; 𝑓q ≤
where σy is the equivalent stress; p is hydrostatic
is𝑓the
effective stress; and q1 , q2 , and q3 are
c
𝑓c +
δ(𝑓 −
𝑓c ) 𝑓c < 𝑓effects
≤𝑓F
(2) equation.
𝑓 ∗ = {of
fitting parameters to improve prediction
void
interaction
in the Gurson damage
𝑓𝐹̅
𝑓 ≥ 𝑓F
The reduction in mechanical strength of the material due to void coalescence is defined by the
where,
damage parameter
f ∗ , expressed as a function of void volume fraction f. For f ∗ = 0, the GTN yield
∗
condition becomes the normal yield condition
that the material has failed. It is
√𝑞112 indicates
𝑓F̅ − 𝑓c and𝑞1f +=
− 𝑞3
(3)
δ=
, 𝑓F̅ =
defined as,
𝑞3
 𝑓F − 𝑓c

f
f ≤ fc

(2)
f∗ =
f c + δ( f − f c ) f c < f ≤ f F


fF
f ≥ fF
where,
f − fc
δ= F
, f =
fF − fc F
q1 +
q
q21 − q3
q3
(3)
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where f c is critical void volume fraction at which void coalescence first occurs, and f F is the void
volume
at which
material
fails at
due
to crack
wherefraction
𝑓c is critical
voidthe
volume
fraction
which
void. propagation.
coalescence first occurs, and 𝑓F is the void
The rate
of increase
inthe
total
void volume
is due to growth of existing voids as well as
volume
fraction
at which
material
fails duefraction
to crackf propagation.
.
The of
rate
of increase
total
void
fractionas,
𝑓̇ is due to growth of existing voids as well as
nucleation
new
voids, fin
. This
canvolume
be described
new
̇ . This can be described as,
nucleation of new voids, 𝑓new
.
.
.
f ̇ = f growth
+ f new
(4)
̇
̇
𝑓 = 𝑓growth + 𝑓new
(4)
The
growth
to the
thehydrostatic
hydrostaticcomponent
component
plastic
strain
The
growthrate
rateofofexisting
existingvoids
voidsis
is proportional
proportional to
ofof
plastic
strain
.p 𝑝
rate
(εkk
by,by,
rate
(ε̇),𝑘𝑘given
), given
.p
f growth = (1 − f )ε 𝑝
(5)
𝑓growth = (1 − 𝑓)ε̇kk
(5)
𝑘𝑘
The nucleation rate of new voids is generally defined by strain-controlled nucleation rule which
The nucleation rate of new voids is generally defined by strain-controlled nucleation rule which
assumes that the voids undergo nucleation as second phase particles which is normally distributed for
assumes that the voids undergo nucleation as second phase particles which is normally distributed
the total particle population. This is described as,
for the total particle population. This is described as,
.
11 ε̅𝑝p − ε𝑁
f𝑓N𝑁
N
𝑝𝑙pl
√
exp
−
f new
=
̇
[− 2 ( S
𝑓new =
)] ε̅ε𝑚m
(6) (6)
S𝑆N𝑁 √2π
2π
2
𝑆𝑁N
where
volume
fraction
of void-nucleating
particles,
ε𝑁 Sand
𝑆𝑁the
are
the
where
f N 𝑓represents
thethe
volume
fraction
of void-nucleating
particles,
and and
ε N and
average
𝑁 represents
N are
average
anddeviation
standard deviation
of the
which particles
voids, respectively.
and
standard
of the strains
atstrains
which at
particles
nucleatenucleate
voids, respectively.
The
steps
involvedininthe
themodified
modified GTN
GTN damage
damage model
is is
illustrated
The
steps
involved
modelusing
usingVUMAT
VUMATsubroutine
subroutine
illustrated
in
the
form
of
a
flowchart
[22]
in
Figure
4.
in the form of a flowchart [22] in Figure 4.
Figure
AZ31 alloy
alloyusing
usingFE
FEapproach
approach[22].
[22].
Figure4.4.Constitutive
Constitutive modeling
modeling of AZ31
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The modified
modifiedGTN
GTNmodel
model
is used
to predict
the strength
characteristics
ofalloy
AZ31
by
The
is used
to predict
the strength
characteristics
of AZ31
byalloy
varying
varying
the
P–B
ratio
(x
1), temperature (x2) and strain rate (x3). The range of tested input parameters
the P–B ratio (x1 ), temperature (x2 ) and strain rate (x3 ). The range of tested input parameters tested are
testedin
are
given
Table
2. The computes
simulationthe
computes
the engineering
stress and engineering
strain,
given
Table
2. in
The
simulation
engineering
stress and engineering
strain, from which
from
which
the
true
stress
and
true
strain
are
calculated
using
classical
mechanics
relations.
the true stress and true strain are calculated using classical mechanics relations.
Table
2. Input
Input parameters
parameters for
for tensile
tensile loading
loading simulation
simulation of
of AZ31
AZ31 alloy.
alloy.
Table 2.
Process Input Variable
Process Input Variable
P–B ratio
P–B ratio
Temperature
Temperature
Strain rate
Strain rate
Values
Values
0.2, 0.3, 0.4, 0.5
0.2,
0.3,400,
0.4, 450
0.5
300, 350,
300, 350, 400, 450
0.0001, 0.001, 0.01
0.0001, 0.001, 0.01
Units
No units
No units
K
K −1
−1 s
Units
s
2.3. Validation of Finite Element Model
2.3. Validation of Finite Element Model
The stress–strain plot of the AZ31 alloy under uni-axial tensile loading is shown in Figure 5. For
The stress–strain
plotwith
of the
AZ31
alloy
uni-axialtensile
tensileloading
loadingcharacteristics
is shown in Figure
5.
comparison
of FE results
actual
data,
theunder
experimental
of AZ31
For
comparison
of [24]
FE results
actualindata,
the experimental
loading
characteristics
of
obtained
from Ref.
are alsowith
included
the figure.
It can be seentensile
from this
plot that
the FE model
AZ31
obtained
from
Ref.
[24]
are
also
included
in
the
figure.
It
can
be
seen
from
this
plot
that
the
FE
was able to predict the stress characteristics of AZ31 with an accuracy of 92%. It is noted that the
model
able toapredict
the high
stressyield
characteristics
of AZ31towith
an accuracy
of 92%.
It attributed
is noted that
model was
predicted
relatively
stress compared
experiments.
This
can be
to
the
model
predicted
highdefects,
yield stress
compared
to experiments.
Thissettings
can be attributed
several
factors
such aasrelatively
the material
temperature
fluctuations,
machine
in tensile
to
several
such asthe
theFE
material
defects,
temperature
fluctuations,
machine
settings in tensile
tester,
etc. factors
Furthermore,
geometry
of AZ31
also factored
the corrosive
film component
which
tester,
etc.
Furthermore,
the
FE
geometry
of
AZ31
also
factored
the
corrosive
film
component
which
may have caused certain variations in the yield stress prediction. This clearly demonstrates the
may
have
caused
certain variations
in the required
yield stress
This
demonstrates
ability
of the
FE simulation
for generating
dataprediction.
sets that can
be clearly
subsequently
utilizedthe
by
ability
of
the
FE
simulation
for
generating
required
data
sets
that
can
be
subsequently
utilized
by
optimization algorithms.
optimization
algorithms.
The tensile
loading simulation performed by ABAQUS/CAE software is depicted in the form of
The
tensile
loading6.simulation
performed
ABAQUS/CAE
is depicted
theas
form
screen shots in Figure
It can be seen
that theby
stress
distribution software
in the AZ31
alloy as in
well
the
of
screen
shots
in
Figure
6.
It
can
be
seen
that
the
stress
distribution
in
the
AZ31
alloy
as
well
as
the
corrosive film depends markedly on the simulation temperature due to the development of thermal
corrosive
thesteps
simulation
temperature
duethe
to tensile
the development
of thermal
stresses infilm
the depends
material. markedly
A total of on
5000
were used
to complete
loading simulation
of
stresses
in
the
material.
A
total
of
5000
steps
were
used
to
complete
the
tensile
loading
simulation
of
the AZ31 material in the CAE model.
the AZ31 material in the CAE model.
Figure 5. Comparison of tensile force of SS304 alloy predicted from FEM simulation with experimental
Figure 5. Comparison of tensile force of SS304 alloy predicted from FEM simulation with
data obtained from Ref. [24].
experimental data obtained from Ref. [24].
Metals 2017, 7, 83
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Metals 2017, 7, 83
8 of 13
(a)
(b)
(c)
(d)
Figure
of mechanical
mechanical stress
stressbuildup
buildupofofcorrosive
corrosiveAZ31
AZ31alloy
alloyatattemperatures:
temperatures:
300
Figure 6.
6. FE
FE modeling
modeling of
(a)(a)
300
K;
K;
(b)
350
K;
(c)
400
K;
and
(d)
450
K.
It
can
be
seen
that,
regardless
of
temperature,
the
stress
(b) 350 K; (c) 400 K; and (d) 450 K. It can be seen that, regardless of temperature, the stress concentration
concentration
is highest
corrosive
film near
pits and
tips which
leads to material
is highest at corrosive
film at
near
pits and crack
tips which
leadscrack
to material
deterioration.
deterioration.
Metals 2017, 7, 83
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3. Analytical Modeling of Stress Corrosion Characteristics of AZ31 Alloy
The FE model generates the strength data of AZ31 alloy by systematically varying the input
parameters related to P–B ratio, temperature and strain rate. A total of 48 data sets are then used to
obtain non-linear analytical model using GP technique. In GP cluster, the training data are used to
train the GP model and the validity of obtained model is tested on testing data. Kennard-and-Stone
algorithm is integrated within the GP algorithm to ensure randomness of data selection for training and
testing sets. The GP cluster is then used to derive mathematical model that describes the relationship
between the true strength of the alloy (y) as a function of input variables, viz. P–B ratio (x1 ), temperature
(x2 ) and strain rate (x3 ). The descriptive statistics of the data sets generated from the FE model is
shown in Table 3.
Table 3. Descriptive statistics of true stress data from FE simulation.
Parameter
P–B Ratio, x1
(No Unit)
Temperature, x2
(K)
Strain Rate, x3
(s−1 )
True Stress, y1
(MPa)
Mean
Median
Standard deviation
Kurtosis
Skewness
Minimum
Maximum
0.35
0.35
0.11
−1.38
−1.842 × 10−15
0.2
0.5
375
375
56.49
−1.38
0
300
450
0.034
0.001
0.047
−1.53
0.730
10−4
0.1
137.43
121.92
49.30
−0.49
0.75
69.82
235
The accuracy and effectiveness of the evolutionary algorithm framework depends on parameter
settings, which control the generation of different sizes of mathematical models. A trial and error
approach is used by the authors to determine the optimum parameter settings in the present study.
The optimal settings determined are listed as follows: the population size is 400, the number of
iterations is 20, the number of maximum genesis 6, the probability for crossover is 0.80, and the number
of generations is 200. The authors deployed the GP algorithm from the GPTIPS toolbox in MATLAB
framework developed by Searson et al. [25]. The best GP model (Equation (7)) for the true stress of
AZ31 alloy (YAZ31 ) is selected based on the minimum mean absolute percentage error value among all
the runs.
YAZ31 (MPa) = 68.18 + (1036.91 × x1 ) + [0.71 × tan{( x2 ) × (log(log( x3 )))}]
−[920.58 × {log(cos(log( x2 )))}] − [ 1130.62 × { tan{(tan( x1 ))}][
+0.53 × { tan(tan(log(log( x3 ))))}] + [0.49 × {x2 × (tan(tan( x1 )))}]
(7)
where x1 , x2 , x3 are values of P–B ratio, temperature (K) and strain rate (s−1 ), respectively. YAZ31 is the
value of true stress derived from the model based on the values of x1 , x2 , x3 .
3.1. Performance Evaluation of the Analytical Model
The evaluation of the FEM-GP model performance is conducted by analyzing the statistical
metrics of the model as defined by the root mean square error (RMSE), the coefficient of determination
(R2 ), and the mean absolute percentage error (MAPE). These metrics provides an indicator of model
performance as they measure the deviations of the predicted from actual values. The definition of
these model metrics represented by mathematical equations is available in Ref. [26]. The performance
and validation analysis of the FEM-GP model is shown in Figure 7. It can be seen from the figure that
the FEM-GP model is able to predict the true stress of AZ31 with close fit and higher accuracy. Hence,
it can be concluded that the FEM-GP model can be used as an effective approach for generalizing the
true stress of AZ31 alloy under corrosion.
Metals 2017, 7, 83
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10 of 13
Metals 2017, 7, 83
10 of 13
(a)
(b)
Figure 7.
7. Performance
true
stress
of of
AZ31
for (a) training data and (b)
Figure
Performanceof
ofFEM-GP
FEM-GPmodel
modeland
andANN
ANNofof
true
stress
AZ31
(a)
(b) for (a) training data and
2: coefficient
testing
data.
R
of
determination;
MAPE:
mean
absolute
percentage
root
2
(b) testing data. R : coefficient of determination; MAPE: mean absolute percentageerror;
error; RMSE:
RMSE: root
Figure
7. error;
Performance
of FEM-GP
model and ANN
of Artificial
true stress Neural
of AZ31 Network.
for (a) training data and (b)
mean
square
GP:
Genetic
Programming;
ANN:
mean square error;2 GP: Genetic Programming; ANN: Artificial Neural Network.
testing data. R : coefficient of determination; MAPE: mean absolute percentage error; RMSE: root
mean square error; GP: Genetic Programming; ANN: Artificial Neural Network.
3.2. Model Analysis Based on Corrosion Physics
3.2. Model Analysis Based on Corrosion Physics
3.2. Model Analysis
Based onisCorrosion
Physics
Corrosion
material
a complex
process
in which
factorsconsiderable
exhibit considerable
Corrosion
ofofmaterial
is a complex
process
in which
severalseveral
factors exhibit
influence
influence Corrosion
on the material
behavior.
Hence, any
analytical
model
describing
the material
strength of
of
material
is
a
complex
process
in
which
several
factors
exhibit
considerable
on the material behavior. Hence, any analytical model describing the material strength of the
the component
corrosion
must be ableany
to capture
the actual
physics the
involved
during
corrosion.
influence
onunder
thecorrosion
material
behavior.
analytical
material
strength
of
component
under
must beHence,
able to capture
the model
actualdescribing
physics involved
during
corrosion.
For this,
parametricunder
analysis
is conducted
which
showsthe
the
variation
ininvolved
the trueduring
stresscorrosion.
based on the
the component
corrosion
must be able
to capture
actual
physics
For this, parametric analysis is conducted which shows the variation in the true stress based on the
For this,input
parametric
analysis
theofvariation
in the true
stress based
on the
considered
factors.
Figureis8conducted
shows thewhich
effectshows
of each
the considered
individual
factors
on the
considered input factors. Figure 8 shows the effect of each of the considered individual factors on the
considered
input
factors.
Figure
8
shows
the
effect
of
each
of
the
considered
individual
factors
onmaterial
the
true stress of AZ31 alloy under corrosion. It can be seen that as the P–B ratio increases the
true stress
of AZ31 alloy
under
corrosion.
It can be
seenthat
thatasasthe
theP–B
P–B ratio
increases
the material
true degrades
stress of AZ31
alloyThe
under
corrosion. Itiscan
be seen
the material
strength
rapidly.
degradation
more
pronounced
whenratio
P–Bincreases
ratio exceeds
0.3. Based
strength
degrades
rapidly.
TheThe
degradation
is is
more
exceeds0.3.
0.3.Based
Based on
strength
degrades
rapidly.
degradation
morepronounced
pronouncedwhen
when P–B
P–B ratio
ratio exceeds
on corrosion physics, it can be stated that the corrosive film does not protect the alloy from
corrosion
physics,
it
can
be
stated
that
the
corrosive
film
does
not
protect
the
alloy
from
degradation.
on corrosion physics, it can be stated that the corrosive film does not protect the alloy from
degradation. Similar trend can be seen for the case of temperature in which a corrosive AZ31 alloy
Similar
trend canSimilar
be seentrend
for the
of temperature
which a corrosive
alloyAZ31
exhibits
degradation.
can case
be seen
for the case ofin
temperature
in which AZ31
a corrosive
alloylower
exhibits
lower
material
strength
at
higher
working
temperatures.
This
indicates
that
AZ31
alloy
exhibits
lowerat
material
higher workingThis
temperatures.
thatunder
AZ31 alloy
material
strength
higherstrength
workingattemperatures.
indicates This
that indicates
AZ31 alloy
corrosive
underunder
corrosive
conditions
is
not
suitable
for
high
temperature
applications.
However,
the
strain
conditions
is not
suitable forapplications.
high temperature
applications.
However,
the to
strain
conditions iscorrosive
not suitable
for high
temperature
However,
the strain
rate seems
have no
rate seems
to have
no no
effect
onon
the
material
strength
of AZ31
AZ31alloy
alloywhich
which
agrees
well
with
previous
rate
seems
to
have
effect
the
material
strength
of
agrees
well
with
previous
effect on the material strength of AZ31 alloy which agrees well with previous experimental study [24].
experimental
study [24].
Hence,
atatambient
temperatures,the
theeffect
effect
loading
strain
experimental
[24].
Hence,
ofof
loading
(i.e.,(i.e.,
strain
rate)rate)
does does
Hence,
at ambientstudy
temperatures,
the ambient
effect oftemperatures,
loading (i.e., strain
rate)
does not
have
any
negative
or
not have
negative
positiveimpact
impacton
on the
the strength
strength ofofthe
alloy.
This
corroborates
with with
not have
any any
negative
or or
positive
theAZ31
AZ31
alloy.
This
corroborates
positive impact on the strength of the AZ31 alloy. This corroborates with the fact that material failure
the that
fact that
material
failure
underSCC
SCCoccurs
occurs not
not because
load
itself,
but but
due due
to theto the
the fact
material
failure
under
becauseofofexternal
external
load
itself,
undercrack
SCCgrowth
occursand
notnucleation
because of
external
load
itself,
butresidual
due tostresses,
the crack
growth
and
nucleation
of
of
voids
that
results
from
loading,
etc.
Hence,
from from
crack growth and nucleation of voids that results from residual stresses, loading, etc. Hence,
voidsthis
thatanalysis
resultsitfrom
residual
stresses,
loading,
etc.
Hence,
from
this
analysis
it
can
be
concluded
that
can be concluded that the derived analytical model was able to confirm well with the
this analysis it can be concluded that the derived analytical model was able to confirm well with the
the derived
analytical
wasprocess.
able to confirm well with the observed physics in corrosion process.
observed
physics inmodel
corrosion
observed physics in corrosion process.
(a)
(a)
(b)
Figure 8. Cont.
(b)
Metals 2017, 7, 83
11 of 13
Metals 2017, 7, 83
11 of 13
(C)
Figure 8. Parametric analysis of true stress of AZ31 with (a) P–B ratio, (b) Temperature and (c) Strain
Figure 8. Parametric analysis of true stress of AZ31 with (a) P–B ratio, (b) Temperature and (c)
rate. Square markers ( ) in these plots refer to the data samples of true stress and the corresponding
Strain rate. Square markers () in these plots refer to the data samples of true stress and the
inputs.
corresponding inputs.
A sensitivity analysis was also conducted to determine the relative importance of each of the
input
parameters
on the was
considered
output. Thetosensitivity
analysis
is defined
by partial derivative
A
sensitivity
analysis
also conducted
determine
the relative
importance
of each of the
strength with
to temperature.
While
partial derivative
thepartial
other inputs
inputofparameters
onrespect
the considered
output.
Thethe
sensitivity
analysisisiscomputed,
defined by
derivative
such as the P–B ratio and strain rate are kept constant at their mean values. The effect of the two
of strength with respect to temperature. While the partial derivative is computed, the other inputs
inputs P–B ratio and strain rate, which are at mean values, is still there on the strength. Therefore,
such as the P–B ratio and strain rate are kept constant at their mean values. The effect of the two
the strength depends on the inputs in a complicated way. The relative importance of the considered
inputs
P–B ratio and strain rate, which are at mean values, is still there on the strength. Therefore, the
input parameters on the mechanical strength of AZ31 alloy under corrosion is depicted in Table 4. It
strength
on the
in a complicated
relative
importance
the temperature,
considered input
can depends
be seen that
the inputs
most important
amongst way.
thoseThe
studied
under
corrosion of
is the
parameters
on
the
mechanical
strength
of
AZ31
alloy
under
corrosion
is
depicted
in
4. It can be
followed by P–B ratio and strain rate. This analysis corroborates with actual corrosion Table
mechanisms
seen in
that
the most alloy
important
those
studied
under
is thetemperatures.
temperature,As
followed
magnesium
system,amongst
as they are
more
susceptible
to corrosion
SCC at elevated
AZ31 by
is primarily
usedrate.
for high
applications,with
controlling
system temperature
crucial
to
P–B ratio
and strain
Thistemperature
analysis corroborates
actual the
corrosion
mechanismsisin
magnesium
tensile
strength
under corrosion,
which
can lead
to material failure.
alloyavoid
system,
as they
aredegradation
more susceptible
to SCC at
elevated
temperatures.
As AZ31 is primarily
used for high temperature applications, controlling the system temperature is crucial to avoid tensile
Table 4. Sensitivity analysis of true stress of AZ31 alloy.
strength degradation under corrosion, which can lead to material failure.
Percentage Contribution
Input Parameter
to stress
True Stress
(%)alloy.
Table 4. Sensitivity analysis of true
of AZ31
P–B ratio
36
Temperature (K) Percentage Contribution
62
Input Parameter
to True Stress (%)
Strain rate (s−1)
2
P–B ratio
36
Temperature (K)
62
4. ConclusionsStrain rate (s−1 )
2
The present work introduced an integrated FE based computational intelligence approach for
modeling
4. Conclusionscorrosion mechanics of AZ31 alloy under mechanical loading. FE modeling of AZ31 alloy
with a modified constitutive damage model was used to predict the true stress responses of AZ31
The
workcorrosive
introduced
an integratedtemperature
FE based computational
intelligence
approach
alloypresent
with varying
film concentration,
and strain rate. The
data obtained
from for
modeling
corrosion
mechanics
of aAZ31
alloy
under mechanical
loading.
modeling
of AZ31
the model
were used
to derive
genetic
programming
model, which
couldFE
predict
the strength
of alloy
alloy with
an accuracy
of 92%model
and confirmed
data trend
withstress
actualresponses
corrosionof
physics.
with athe
modified
constitutive
damage
was used the
to predict
the true
AZ31 alloy
and parametric
analysis of the genetic
programming
model
showed
that the
mechanical
with Sensitivity
varying corrosive
film concentration,
temperature
and strain
rate.
The data
obtained
from the
strength
of
the
AZ31
alloy
highly
dependent
on
system
temperature
and
does
not
show
much
model were used to derive a genetic programming model, which could predict the strength of
the alloy
with regards to applied strain rate. The proposed model can be used to determine optimal
with variation
an accuracy
of 92% and confirmed the data trend with actual corrosion physics. Sensitivity and
design parameters for AZ31 alloy under corrosive environments without the need of experiments.
parametric analysis of the genetic programming model showed that the mechanical strength of the
This can help in reducing costs associated with material fabrication and testing, which can also lead
AZ31toalloy
highly dependent on system temperature and does not show much variation with regards
sustainable materials design.
to applied strain rate. The proposed model can be used to determine optimal design parameters
for AZ31 alloy under corrosive environments without the need of experiments. This can help in
reducing costs associated with material fabrication and testing, which can also lead to sustainable
materials design.
Metals 2017, 7, 83
12 of 13
Author Contributions: The work presented here was carried out in collaboration between all authors.
Venkatesh Vijayaraghavan and Akhil Garg defined the research theme. Venkatesh Vijayaraghavan performed
the finite element modeling and wrote the paper. Akhil Garg performed the genetic programming computations.
Liang Gao provided the genetic programming framework and provided guidance on interpreting the trends from
the evolutionary model. Rangarajan Vijayaraghavan designed methods and experiments, analyzed the data, and
interpreted the results. All authors have contributed to, seen and approved the manuscript. The authors hope that
this paper can contribute to the successful application of finite element modeling to determine optimal design
parameters of AZ31 alloy without the need to conduct experiments.
Conflicts of Interest: The authors declare no conflict of interest.
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© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access
article distributed under the terms and conditions of the Creative Commons Attribution
(CC BY) license (http://creativecommons.org/licenses/by/4.0/).