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j o u r n a l o f t h e m e c h a n i c a l b e h a v i o r o f b i o m e d i c a l m a t e r i a l s 5 3 ( 2 0 1 6 ) 3 5 0 – 3 6 5
http://dx.doi.org/10.1751-6161/& 2015 El
nCorresponding aE-mail address:
Imprecise knowledge based design and developmentof titanium alloys for prosthetic applications
S. Dattaa,n, M. Mahfoufb, Q. Zhangc, P.P. Chattopadhyayd, N. Sultanae
aB.U. Institute of Engineering Bankura 722146, IndiabDepartment of Automatic Control and Systems Engineering The University of Sheffield, Sheffield S1 3JD, UKcSchool of Engineering and Digital Arts University of Kent, Canterbury CT2 7NT, UKdNational Institute of Foundry and Forge Technology Hatia 834003, Ranchi, IndiaeSchool of Materials Science and Engineering Indian Institute of Engineering Science and Technology, Shibpur,Howrah 711103, India
a r t i c l e i n f o
Article history:
Received 24 May 2015
Received in revised form
23 August 2015
Accepted 29 August 2015
Available online 8 September 2015
Keywords:
Titanium alloy
Orthopedic application
Alloy design
Rule based modeling
Evolutionary algorithm
Multi-objective optimization
1016/j.jmbbm.2015.08.039sevier Ltd. All rights rese
uthor. Tel.: þ91 3242 [email protected] (S. D
a b s t r a c t
Imprecise knowledge on the composition–processing–microstructure–property correlation
of titanium alloys combined with experimental data are used for developing rule based
models for predicting the strength and elastic modulus of titanium alloys. The developed
models are used for designing alloys suitable for orthopedic and dental applications.
Reduced Space Searching Algorithm is employed for the multi-objective optimization to
find composition, processing and microstructure of titanium alloys suitable for orthopedic
applications. The conflicting requirements attributes of the alloys for this particular
purpose are high strength with low elastic modulus, along with adequate biocompatibility
and low costs. The ‘Pareto’ solutions developed through multi-objective optimization show
that the preferred compositions for the fulfilling the above objectives lead to β or near β-
alloys. The concept of decision making employed on the solutions leads to some
compositions, which should provide better combination of the required attributes. The
experimental development of some of the alloys has been carried out as guided by the
model-based design methodology presented in this research. Primary characterizations of
the alloys show encouraging results in terms of the mechanical properties.
& 2015 Elsevier Ltd. All rights reserved.
rved.
60.atta).
1. Introduction
The metallic biomaterials are considered the most suitable
for replacing failed hard tissues, and among them titanium
alloys are getting much attention due to excellent specific
strength and the best biocompatibility. Pure titanium and Ti-
6Al-4V still occupy most of the market of titanium biomater-
ials (Niinomi, 2003; Titanium Alloy Guide). But due to lower
elastic modulus, β-type titanium alloys are found advanta-geous from the point of mechanical biocompatibility,regarded as an important factor. This property is importantas a low elastic modulus reduces stress shielding of theneighboring bone (Mohammed et al., 2014; Kim et al., 2008;CHOE et al., 2009). But pure Ti and αþβ type Ti-6Al-4V alloyshave an elastic modulus of 105–110 GPa, while the humancortical bone has a stiffness of around 20 GPa (Hin, 2004;Oldani and Dominguez, 2012). β or near-β alloys include Mo,
j o u r n a l o f t h e m e c h a n i c a l b e h a v i o r o f b i o m e d i c a l m a t e r i a l s 5 3 ( 2 0 1 6 ) 3 5 0 – 3 6 5 351
Nb, Zr, Fe, Ta, Pd and Sn as alloying elements (Joshi;Eisenbarth et al., 2004). The implant materials also need highfatigue strength in addition to the low elastic modulus andhigh strength. Therefore, alloying elements such as Cu, Co,Ni, and Si are used for the purpose of strengthening (Polmear,2006). The β-phase being highly plastic, forming of β-Ti-alloysbecomes easy (Weiss and Semiatin., 1998). In addition, somerecent concern has been highlighted over the use of Ti-6Al-4V, as it seems that Al and V dissolve out inside human bodyand may have a negative effect on the body. Thus, substitu-tion of V by Nb or Ta, and substitution of Al by Zr can beadvantageous from this aspect also (Li et al., 2014; Jae-Samet al., 2008; Nasab and Hassan, 2010).
It seems from the above that a judicious selection ofcomposition may lead to further improvement of the propertiesof the Ti alloys suitable for its use as an implant material. Thusthe information generated by previous workers through experi-mental findings (Niinomi, 2003; Mohammed et al., 2014; Ozakiet al., 2004; Kulkarni et al., Iglič; Gonza' lez et al., 2009; Gepreeland Niinomi) is utilized in this research in addition to the priorknowledge of the system to find the optimum composition of Tialloys to achieve the best combination of high strength, lowelastic modulus, adequate biocompatibility and with low costs.But searching the optimum combination of the alloying addi-tions in the experimental domain becomes expensive from theview point of time andmoney. On the other hand computationaldesign strategies have the capacity to stimulate ‘right first time’experimentation (Mahfouf et al., 2009; Zhang et al., 2011), whichaims to achieve the targeted properties within a few attempts.The ‘first principle’ techniques are utilized by several workersusing parameter-free density functional theory calculations toprovide guidance in selecting and optimizing Ti alloys withrespect to the use of non-toxic alloy elements, the stabilizationof the β-phase and the reduction of the stiffness compared(Raabe et al., 2007) or by means of ab initio plane-wave pseudo-potential method to investigate mono-vacancy, di-vacancy andself-interstitials in α titanium (Raji et al., 2009). There are severalattempts to develop composition-property correlation in othermaterials also, e.g. single crystal elastic constants of Mg–Li alloysusing density functional theory (DFT) (Counts et al., 2009),semiconductor spintronics (Katayama-Yoshida and Sato, 2003),and Pb(Sc0.5Nb0.5)O3 perovskite alloy (Íñiguez and Bellaiche,2001), etc. However, informatics-based design using computa-tional intelligence techniques have been successfully used fordesigning Ti alloys (Datta et al., 2013; Sultana et al., 2014) andother materials (Datta and Chattopadhyay, 2013; Dey et al., 2008;Nandi et al., 2012). Informatics based materials design strategiestry to extract the inherent correlation within data and utilize theinformation for designing novel materials (Rajan, 2005). Statis-tical as well as computational intelligence based techniques havethe capacity to use data as well as imprecise knowledge (expert-knowledge) of a system and derive a model of the system thathas the capability to generalize behavior over a wide range ofparameter variability. It can thus successfully and specificallyhandle noise as well as uncertainties in complex materialssystems (Bhadeshia, 2008), such techniques are known to be‘universal approximators’. Such efforts ultimately reinforce the‘first principle’ approaches of materials modeling and design. Inthis work models for yield strength and elastic modulus of Tialloys are developed using an innovative two-layered fuzzy
inference system (FIS), where the first layered correlating thecomposition and process variables with the microstructuralfeatures is developed using prior-knowledge, and the secondlayer correlating the microstructure with properties were devel-oped using published experimental data. An fuzzy inferencesystem is a modeling tool based on fuzzy logic (Zadeh, 1973),which can be defined as an extension of multivalued logic. It hasits origin in the theory of fuzzy sets, a theory which relates toclasses of objects with transitional boundaries in which amembership of the set is a matter of ‘degree.’ FIS can useimprecise knowledge of a system in the form of if-then rules tomap the relationship between inputs and outputs (Mamdani andAssilian, 1975). Fuzzy logic can also be used to develop if-thenrules from experimental data, which can be used to develop themodel. Such models can extract useful knowledge about thesystem, particularly that relating the microstructural featureswith the properties, in the form of if-then statements. Thedeveloped models are then used for identifying optimal combi-nations of compositions and processing for high strength andlow elastic modulus. The objectives being conflicting in nature,multi-objective optimization is employed using an evolutionaryalgorithm known as Reduced Space Searching Algorithm (RSSA)(Zhang and Mahfouf, 2007, 2010). Evolutionary algorithm is aterm used to describe computer based problem solving systemswhich use computational models of evolutionary processes askey elements in their design and implementation (Deb, 1995).RSSA is one type of evolutionary algorithm that is inspired byhumans' natural searching behavior. To seek a target within alarge and complex search space, a general means to reducing theworkload could be of dividing the search space into parts andstrategically arranging the sequence and density of searchingamong the various subspaces, as shown in Fig. 1. RSSA employssuch an idea hierarchically and repeatedly so as to achieve thefast convergence and accurate optimization.
It is worth noting that the biocompatibility issue is treatedas a constraint in the optimization process. For costs of aweighted summation model of the elemental constituent isdeveloped with the recent cost of the alloying additions. Incase of multi-objective optimization, unlike the single objec-tive optimization, a set of non-dominated solutions, whereeach solution is better than other in at least one objective,evolve;this set of solutions is called Pareto solutions(Goldberg, 2002; Chakraborti, 2004; Deb, 2001). In the presentstudy, the compositions of alloys selected from the evolvedPareto solutions are developed for experimental trials.
2. Database
The published experimental data are collected from varioussources (Niinomi, 2003; Kuroda et al., 1998; Davidson and Kovacs,1992; ⟨http://www.feppd.org/ICBDent/campus/biomechanics_in_dentistry/ldv_data/mech/basic_bone.htm⟩; Schneidera et al., 2005;Niinomi, 2003; Wang et al., 1989; ⟨http://www.efunda.com/materials/alloys/titanium/properties.cfm⟩; Donachie, 2000;Schneider et al., 2000; He and Hagiwara, 2006). The input variablesconsist of compositional variables and processing parameters andthe output variable are themechanical properties. For strength, thenumber of training data is 145 and the number of testing data is26, and for elastic modulus, the number of training data is 165 and
Fig. 1 – The Reduced Space Searching strategy for optimization.
Table 1 – List of input and output variables with theirlower and upper limits.
S.No. Variables Lower limit Upper limit
1 Aluminum (Al) 0 82 Vanadium (V) 0 153 Zirconium (Zr) 0 184 Molybdenum (Mo) 0 155 Niobium (Nb) 0 41.16 Tantalum (Ta) 0 307 Iron (Fe) 0 2.58 Tin (Sn) 0 159 Silicon (Si) 0 0.5
10 Chromium (Cr) 0 2011 Testing temperature 25 85012 Solutionising temperature (Tsoln) 718 100013 Ageing temperature (Tage) 25 60014 Ageing time (tage) 0 432015 Cooling rate 0.5 1001 Elastic modulus (EM) 46 1252 Yield strength (YS) 250 1317
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the number of testing data is 29. The list of input and outputvariables, with their minimum andmaximum values, is presentedin Table 1.
3. Fuzzy model development
A Fuzzy Inference System (FIS) that maps an input space intoan output space using the theory of fuzzy logic (Zadeh, 1973;Mamdani and Assilian, 1975) is used in this work. The FIS hasthree parts, viz. fuzzy membership functions, fuzzy operatorsand specific implication operations in the form of if-thenstatements or rules to give a quantitative meaning to aqualitative (linguistic) phrase (statement). The membershipfunction of a fuzzy set represents the degree of truth, wherefuzzy truth represents membership in vaguely defined sets. Thefuzzy operators are the familiar operators like AND, OR, andNOT, can expressed as intersection, union, and complement.This non-linear mapping is capable of dealing with ambiguities
j o u r n a l o f t h e m e c h a n i c a l b e h a v i o r o f b i o m e d i c a l m a t e r i a l s 5 3 ( 2 0 1 6 ) 3 5 0 – 3 6 5 353
(uncertainties) surrounding the real-world and is also able todeal with what is known as ‘the curse of dimensionality’. Asample fuzzy rule can be expressed as follows:
If, P is X and Q is Y then R is Z.where X, Y and Z are linguistic variables (labels) defined by
fuzzy sets on the specific ranges e.g. high, low, fast slow, etc.,while P and Q are the input variables, and R is the outputvariable. In such rules the ‘if’ part is called the “antecedent”and the ‘then’ part is called the “consequent”. In the abovefuzzy rule, the inputs and outputs would be both representedby fuzzy sets in case of Mamdani-type representation(Mamdani and Assilian, 1975). The FIS is explained in Fig. 2.As the Mamdani-type FIS is used in the present work, othertype of FISs are not discussed.
In this work, predefined fuzzy sets are generated using theknowledge of experts (the authors). The rule-bases for thefirst layer of the models, i.e. the rules correlating the compo-sition and microstructural features, are developed from priorknowledge. In case of the second layer correlating the micro-structural features to the mechanical properties, the primarywork consists of generating appropriate fuzzy rule-base. Anyredundancy of the fuzzy rule is estimated via the following‘confidence’ measure (Ishibuchi et al., 2001):
A AA
xx
B C C BC
Ν μ yΝ
conf( → ) = ( ) × ( )( )
=∑ ( ( ) × ( ))
∑ ( ) (1)A
A
nm
n B n
nm
n
=1
=1
where |C(A)�C(B)| is the number of training patterns compa-tible to the antecedent (the ‘if’ part of the rule) A and the
Fuzzification
Crisp inputs
Inference
Rule base
Database Knowledge base
Defuzzification
Crisp output
Fig. 2 – Schematic of Mamdani Fuzzy Inference System.
Molybdenum-equi
Aluminium
Zirconium
Tin
Deformation Temp.
Solutionising Temp.
Cooling Rate
Ageing Temp.
Ageing Time
Volume F
Solution
Grain Siz
Aciculari
Dislocati
Decomp
Fig. 3 – Schematic of the fuzzy models (compositio
consequent (the ‘then’ part of the rule) B, where as |C(A)| is thesame for the antecedent A only. NA(xn) is the grade ofcompatibility of the input xn along with antecedent A¼ [A1,A2, …, Ai]
T, and μB(yn) is that of the output value yn with theconsequent B. NA(xn) is defined by the minimum or theproduct operator, such that:
( )xΝ μ x μ x μ x( ) = min ( ), ( ), ... , ( ) (2)A n A n A n A in1 2j j ij1 2
or
xΝ μ x μ x μ x( ) = ( ) × ( ) × ... × ( ) (3)A n A n A n A in1 2j j ij1 2
In this case, to reduce the complexity of the system, andfor the ease of expressing the existing domain knowledge inthe form of if-then rules, the number of inputs is reduced fromfourteen (14) to nine (9), using Mo-equivalent (Polmear, 2006),where,
Mo−equivalent = Mo + 0.6*V + 0.28*Nb + 0.22*Ta + 1.25*Cr
+ 2.5*Fe (4)
Thus, the nine variables used for modeling are Mo-equivalent, Al, Zr, Sn, deformation temperature, solutionisingtemperature, cooling rate, ageing temperature and ageingtime. Two separate models are developed for two outputs, i.e.yield strength (YS) and elastic modulus (EM), as per theschematics shown in Fig. 3.
Six Mamdani FISs correlating the composition and processvariables with microstructural features are developed usingprior knowledge of the system. Two Mamdani FISs correlatingthe microstructural features with the properties is developedusing the database generated from published literatures.
3.1. FIS for volume fraction of β
Among the alloy additions most of the elements are β
stabilizers, whose individual effects are combined in the formof Mo equivalent (Kuroda et al., 1998). Al is α stabilizer. Highersolutionising temperature and faster cooling rate leads to
raction of β
Strengthening
e
ty
on Density
osition Products
Yield Strength
Elastic Modulus
n-process-microstructure-property correlation).
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higher volume fraction of β. This concept is utilized to
develop the rule base for this FIS. The rules are as follows:
1. If (Mo equivalent is high) and (Al content is low) and(solutionising temp is high) and (cooling rate is fast) then
(volume fraction of β is high)2. If (Mo equivalent is low) and (Al content is high) and
(solutionising temp is low) and (cooling rate is slow) then
(volume fraction of β is low)3. If (Mo equivalent is low) and (Al content is high) and
(solutionising temp is medium) and (cooling rate is mod-
erate) then (volume fraction of β is low)4. If (Mo equivalent is medium) and (Al content is medium)
and (solutionising temp is medium) and (cooling rate is
moderate) then (volume fraction of β is medium)5. If (Mo equivalent is high) and (Al content is medium) and
(solutionising temp is high) and (cooling rate is moderate)
then (volume fraction of β is medium)6. If (Mo equivalent is high) and (Al content is low) and
(solutionising temp is high) and (cooling rate is moderate)
then (volume fraction of β is high)
The surface view genrated by the developed FIS shows how
much prior imprecise knowledge the model can capture (Fig. 4).
0
5
10
15
20
25
01
23
45
67
8
0.6
0.7
0.8
Mo equivalent, w
t%
Al, wt%
vol f
ract
bet
a
2040
6080
1000.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
Cooling rt, deg C/s
Vol
frac
t bet
a
Fig. 4 – Surface views generated by the FIS describing the effectsMo equivalent and (c) cooling rate and Mo equivalent on volum
3.2. FIS for solid solution hardening
Here, the variation of hardness due to addition of alloying
elements, found by previous workers (Cui and Guo, 2009), is
considered to create the rule base as well as to fix the scale.
Among the alloying elements the purpose of adding Zr is chiefly
due to its solid solution hardening, as it is a neutral element
from the point of phase stabilization (Polmear, 2006). Other
than that, Al, Sn and all other alloying additions (β stabilizers)
have some effect on the final properties. All these effects are
summarized in the form of if-then rules. The role of the β
stabilizers is described in the form of Mo equivalent. The effect
of solid solution hardening is expressed in terms of Hardness
(soln) (VHN). The rules generated for the FIS are as follows:
1. If (Mo equivalent is high) and (Al content is high) and (Zrcontent is high) and (Sn content is high) then (solution
hardening is high).2. If (Mo equivalent is low) and (Al content is low) and (Zr content
is low) and (Sn content is low) then (solution hardnes is low).3. If (Mo equivalent is high) and (Al content is high) and (Zr
content is medium) and (Sn content is medium) then
(solution hardness is medium).4. If (Mo equivalent is medium) and (Al content is medium)
and (Zr content is low) and (Sn content is low) then
(solution hardnes is low).
05
1015
2025
700
750
800
850
0.5
0.55
0.6
0.65
Mo equivalent, wt%
Soln temp, deg C
vol f
ract
bet
a
05
1015
2025
Mo equivalent, wt%
of (a) Al and Mo equivalent, (b) solutionising temperature ande fraction of β.
05
1015
2025
02
4
68
180
200
220
240
Mo equivalent, wt%Al, wt%
hard
ness
(sol
n), V
HN
05
1015
2025
0
5
10
15
180
190
200
210
220
230
240
250
Mo equivalent, wt%Zr, wt%
Har
dnes
s (s
oln)
, VH
N
05
1015
2025
0
5
10
15
180
190
200
210
220
230
240
250
Mo equivalent, wt%Sn, wt%
Har
dnes
s (s
oln)
, VH
N
Fig. 5 – Surface views generated by the FIS describing the effects of (a) Al and Mo equivalent, (b) Zr and Mo equivalent and(c) Sn and Mo equivalent on solid solution hardening.
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5. If (Mo equivalent is high) and (Al content is medium) and(Zr content is high) and (Sn content is high) then (solution
hardness is high).
The surface views of the FIS are shown in Fig. 5.
3.2.1. FIS: grain sizeMost of the β stabilizers act as grain refiners, by increasing the
recrystallization temperature. Hence, the Mo equivalent was
used as a qualitative expression for the grain refinement due
to alloying addition. Daisuke Kuroda et al. observed that the
average grain size of the alloys varies with varying amounts
of Mo, Nb, Zr, Sn and other alloying elements (Weiss and
Semiatin., 1998; Li et al., 2014). A lower deformation tempera-
ture increases the grain refinement as well as defects in the
structure. A lower solution temperature reduces the possibi-
lity of grain growth. These factors are expressed in terms of
fuzzy if-then rules as follows:
1. If (Mo equivalent is low) and (deformation temperature ishigh) and (solutionising temperature is high) then (grain
size is high).
2. If (Mo equivalent is high) and (deformation temperature islow) and (solutionising temperature is low) then (grain sizeis low).
3. If (Mo equivalent is medium) and (deformation tempera-ture is medium) and (solutionising temperature is med-ium) then (grain size is medium).
4. If (Mo equivalent is low) and (deformation temperature islow) and (solutionising temperature is low) then (grain sizeis medium).
5. If (Mo equivalent is medium) and (deformation tempera-ture is low) and (solutionising temperature is low) then(grain size is low).
The surfaced generated from the FIS prediction are shownin Fig. 6.
3.2.2. FIS: acicularityThe acicularity of microstructure has significant effect on themechanical properties of the Ti alloys. The acicular structuresmay form due to transformation at low temperature (mar-tensite formation) or due to formation of Wüdmanstattenstructure. Though unlike steel, martensites in Ti alloys arenot strong, this factor was incorporated in the microstructureto explain its specific role in strength and elastic modulus.
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Filip et al. (2003) noticed that the thickness and length of the
α-phase decrease with increasing cooling rate and with
increasing content of the β -stabilising elements. Acicularity
is expressed here in terms of average aspect ratio of the
grains. The rules generated from the concept of physical
metallurgy of Ti alloys are as follows:
05
1015
2025 100 200 300 400 500 600 700 800
20
25
30
35
40
45
50
Deform temp, deg C
Mo equivalent, wt%
Gra
in s
ize,
mic
ro m
Fig. 6 – Surface views generated by the FIS describing the effect(b) solutionising temperature and deformation temperature on g
05
1015
2025
0
5
10
15
8.2
8.25
8.3
8.35
8.4
8.45
Mo equivalent, wt%Zr, wt%
Asp
ect r
atio
7.8
8
8.2
8.4
Asp
ect r
atio
2040
6080
1004
6
8
10
12
Cooling rt, degC/s
Asp
ect r
atio
Fig. 7 – Surface views generated by the FIS describing the effect(c) cooling rate and solutionising temperature on the acicularity
1. If (Mo equivalent is high) and (Zr content is low) and (Sncontent is low) and (solutionising temperature is high) and
(cooling rate is fast) then (aspect ratio is high).2. If (Mo equivalent is low) and (Zr content is low) and (Sn
content is low) and solutionising temperature is high) and
(cooling rate is slow) then (aspect ratio is high).3. If (Mo equivalent is high) and (Zr content is high) and (Sn
content is high) then (aspect ratio is low).
100200
300400
500600
700800
700
750
800
850
25
30
35
40
45
50
Deform temp, deg C
Soln temp, deg C
Gra
in s
ize,
mic
ro m
s of (a) Mo equivalent and deformation temperature, andrain size.
05
1015
2025
0
5
10
15Mo equivalent, wt%
Sn, wt%
700750
800850
Soln temp, deg C
s of (a) Zr and Mo equivalent, (b) Sn and Mo equivalent, andof microstructure.
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4. If (Mo equivalent is low) and (Zr content is low) and (Sncontent is low) and (solutionising temperature is low) and(cooling rate is slow) then (aspect ratio is low).
5. If (Mo equivalent is medium) and (Zr content is medium)and (Sn content is medium) and (solutionising temperatureis medium) and (cooling rate is moderate) then (aspectratio is medium).
Fig. 7 shows the surfaces generated by the FIS.
3.2.3. FIS: defect in the microstructureDefect in the structure is generally shown through thedislocation density. A higher density of dislocation is gener-ated by deformation below the recrystallization temperature,lower solutionising temperature and faster cooling. Here theconcepts used are not specific to Ti alloys. The rules for thisFIS are as follows:
1. If (deformation temperature is low) and (solutionisingtemperature is low) and (cooling rate is fast) then (disloca-tion density is high).
2. If (deformation temperature is high) and (solutionisingtemperature is high) and (cooling rate is slow) then (dis-location density is low).
3. If (deformation temperature is medium) and (solutionisingtemperature is medium) and (cooling rate is moderate)then (dislocation density is medium).
4. If (deformation temperature is low) and (solutionisingtemperature is medium) and (cooling rate is slow) then(dislocation density is low).
The surface views generated as shown in Fig. 8.
3.2.4. FIS: other decomposition productsThe solutionised alloy, when aged, forms several types ofintermetallics as precipitates within the matrix; this leads toprecipitation hardening. Ageing at 350 1C hard and brittle ω
phase precipitates in a dispersive manner, which leads to thesignificant increase of elastic modulus along with decrease ofplasticity (Polmear, 2006; Cui and Guo, 2009). But when thealloy is aged above 350 1C the ω precipitation is reduced withthe appearance of dot α precipitation, which decrease elasticmodulus and strength and increase plasticity. Here, it isexpressed in terms of Hardness (pptn) (VHN). A method
100200300400500600700800
700750
800850
260
280
300
320
340
Deform temp, oCSoln temp, oC
200
250
300
350
Dis
l den
sity
, 1010
/cm
2
Dis
l den
sity
, 1010
/cm
2
Fig. 8 – Surface views generated by the FIS describing the effect(b) solutionising temperature and cooling rate on the dislocation
similar to that employed in the case of solid solution hard-ening is employed here also. The formulated fuzzy rules areas follows:
1. If (Mo equivalent is medium) and (solutionising temp ishigh) and (cooling rate is fast) and (ageing temp is medium)and (ageing time is medium) then (hardness_pptn is high).
2. If (Mo equivalent is high) and (solutionising temp ismedium) and (cooling rate is fast) and (ageing temp ishigh) and (Ageing time is high) then (hardness_pptn ishigh).
3. If (Mo equivalent is low) and (solutionising temp is low)and (cooling rate is slow) and (ageing temp is low) and(Ageing time is low) then (hardness pptn is low).
4. If (Mo equivalent is high) and (solutionising temp is high)and (cooling rate is slow) and (ageing temp is low) and(ageing time is low) then (hardness pptn is low).
5. If (Mo equivalent is medium) and (solutionising temp ismedium) and (cooling rate is moderate) and (ageing tempis high) and (ageing time is high) then (hardness_pptn ismedium).
The surface views describes the effects of the variables onprecipitation hardening, as depicted by the rule based FIS(Fig. 9).
The first part of the model, correlating the compositionand process variables with the microstructural features, isgenerated from prior knowledge as described above. Thismodel consisting of six FISs is same for both models, i.e.models for yield strength and elastic modulus. The secondpart of the models is generated separately using data col-lected from several published literatures as stated above.Here, the FIS is generated through rules which are evolvedusing the error minimization for predicting the existingdatabase using all possible rule sets. The optimization usesthe Reduced Space Searching Algorithm (RSSA) (Zhang andMahfouf, 2007, 2010); this algorithm is a nature-inspiredsearch technique which aims to shift the search space to asubspace including the ‘optimum’. In this technique, noequations (e.g. derivative-based equations) or evolutionaryoperators (such as mutation and crossover) are used, thuscreating the opportunity to integrate this idea with otheroptimization techniques. This algorithm has already beensuccessfully validated by a number of industrial applications
700750
800850
2040
6080
100
Cooling rt, o C/s
Soln temp, oC
s of (a) solutionising and deformation temperatures, anddensity in the microstructure.
05
1015
2025
2040
6080
100360
380
400
420
440
460
Mo equivalent, wt%Cooling rt, oC/s
Har
dnes
s (p
ptn)
, VH
N
700750
800850
2040
6080
100
360
380
400
420
440
460
Soln temp, oC
Cooling rt, oC/s
Har
dnes
s (p
ptn)
, VH
N
050
100150
200250
300350
100200
300400
500
360
380
400
420
440
460
Ageing time, sAgeing temp, oC
Har
dnes
s (p
ptn)
, VH
N
Fig. 9 – Surface views generated by the FIS describing the effects of (a) cooling rate and Mo equivalent, (b) cooling rate andsolutionising temperature, and (c) ageing temperature and ageing time on precipitation hardening.
j o u r n a l o f t h e m e c h a n i c a l b e h a v i o r o f b i o m e d i c a l m a t e r i a l s 5 3 ( 2 0 1 6 ) 3 5 0 – 3 6 5358
(Zhang and Mahfouf, 2010). The rules, generated in this
process for the two FISs correlating the microstructure with
two properties, can be used for knowledge elicitation, since
the correlation between microstructure and mechanical
properties, particularly in case of elastic modulus, is not
clearly known.
3.2.5. FIS: yield strengthAs stated above the rules relating the microstructural fea-
tures and the YS for Ti alloys are extracted from the database
using optimization technique. The rules generated from the
minimization of the error for predicting the existing database
are as follows:
1. If (volume fraction of β is low) and (solution hardening islow) and (grain size is low) and (acicularity is high) and
(dislocation density is high) and (decomposition product is
low) then (yield strength is high).2. If (volume fraction of β is low) and (solution hardening is
medium) and (grain size is medium) and (dislocation
density is high) and (decomposition product is medium)
then (yield strength is high).3. If (solution hardening is medium) and (grain size is low)
and (acicularity is high) and (dislocation density is high)
and (decomposition product is medium) then (yieldstrength is medium).
4. If (solution hardening is medium) and (acicularity ismedium) and (dislocation density is high) and (decom-position product is low) then (yield strength is low-medium).
5. If (volume fraction of β is medium) and (solution hard-ening is medium) and (grain size is high) and (acicularityis low) and (dislocation density is low) then (yield strengthis high-medium).
6. If (volume fraction of β is medium) and (grain size is high)and (acicularity is low) and (dislocation density is high)and (decomposition product is low) then (yield strength ishigh-medium).
7. If (volume fraction of β is low) and (grain size is high) and(dislocation density is medium) then (yield strength islow).
8. If (volume fraction of β is low) and (solution hardening ismedium) and (grain size is high) and (dislocation densityis low) and (decomposition product is high) then (yieldytrength is low-medium).
9. If (volume fraction of β is low) and (solution hardening ismedium) and (grain size is high) and (acicularity ismedium) and (decomposition product is high) then (yieldstrength is medium).
j o u r n a l o f t h e m e c h a n i c a l b e h a v i o r o f b i o m e d i c a l m a t e r i a l s 5 3 ( 2 0 1 6 ) 3 5 0 – 3 6 5 359
10. If (solution strength is high) and (grain size is low) and(acicularity is high) and (dislocation density is low) and(decomposition product is high) then (yield strength ishigh-medium).
11. If (volume fraction of β is low) and (grain size is medium)and (acicularity is high) and (dislocation density is med-ium) and (decomposition product is high) then (yieldstrength is low-medium).
12. If (volume fraction of β is medium) and (solution hardening islow) and (grain size is high) and (acicularity is low) and(dislocation density is low) then (yield strength is high).
13. If (volume fraction of β is low) and (solution hardening islow) and (grain size is high) and (acicularity is high) then(yield strength is medium).
14. If (volume fraction of β is low) and (solution hardening islow) and (grain size is medium) and (decompositionproduct is high) then (yield strength is high-medium).
15. If (volume fraction β is high) and (solution hardening ismedium) and (grain size is low) and (dislocation density ishigh) and (decomposition product is low) then (yieldstrength is low).
The high number of elicited rules indicates the complexityof the system. But this also makes it difficult to understandthe correlation described through the linguistic expressions.Still an in-depth study of the rules shows that higher amountof β reduces the strength. But it is difficult to understand thecomplex contribution of decomposition products of β, devel-oped through ageing. It is worth noting here that the β grainsmay contain α, α′′and ω phases, depending on the composi-tion and processing (Polmear, 2006). But the roles of thesephases vary significantly on the final property of the alloy. Onthe one hand the, α and ω phases increase the strength andelastic modulus, with the effect of ω more profound than theother one. On the other hand α′′ has the reverse such a role.Thus it becomes difficult for the rules to capture the phe-nomena in a simpler way. A similar complexity exists in caseof solution hardening, grain size and defects in microstruc-ture. The real relations are better revealed by the surfaceplots developed from the FIS guided by the above rules, whichare reported later. The model predictions for the existing data
200 400 600 800 1000 1200
200
400
600
800
1000
1200
Measured Outputs
Pre
dict
ed O
utpu
ts
Fig. 10 – Scatter plot showing model predicted versusmeasured yield strength values.
are shown in Fig. 10. The excellent predictability of the novelsemi-empirical model developed using imprecise knowledgeas well as data shows that how imprecise knowledge of amaterials system could be utilized to develop a predictivemodel.
The surface views generated by the model are shown inFig. 11. Here, it can be seen that the effect of volume fractionof β has been clearly depicted by the model prediction, buteffect of grain size is not that significant. In case of solidsolution hardening it can be seen that initially the strength isincreased, but then it diminishes (Fig. 11b). This may be dueto fact that here most of the alloying additions are β
stabilizers, and achieving higher solution hardening meansa higher alloying addition, which leads to increase in β
content. This in turn leads to a decrease in strength. Theeffect of precipitates is not profound at low solution hard-ening, i.e. in the low alloying addition regime, i.e. in the α
phase. But the situation becomes complex in the β phasedomain. The role of defects seems to be not so significant(Fig. 11c). In the case of a data base where cold deformation ismissing, and mostly there exists some post deformation heattreatment, such behavior is expected. The role of aspect ratiois rather complex and it can be seen that in the medium levelit leads to softening.
3.2.6. FIS: elastic modulusThe rules for the elastic modulus model are elicited using thesame method as for the yield strength model, and are asfollows:
1. If (volume fraction of β is low) and (solution hardening ismedium) and (grain size is low) and (acicularity is med-ium) and (dislocation density is high) and (decompositionproduct is high) then (elastic modulus is low).
2. If (solution strength is high) and (grain size is medium) and(acicularity is high) and (dislocation density is low) and(decomposition product is low) then (elastic modulus ismedium).
3. If (volume fraction of β is high) and (solution hardening islow) and (grain size is low) and (acicularity is high) and(dislocation density is low) and (decomposition product islow) then (elastic modulus is low).
4. If (grain size is low) and (acicularity is high) and (disloca-tion density is low) and (decomposition product is low)then (elastic modulus is low-medium).
5. If (volume fraction of β is medium) and (solution hard-ening is low) and (grain size is medium) and (acicularity islow) and (decomposition product is medium) then (elasticmodulus is high-medium).
6. If (volume fraction of β is high) and (solution hardening islow) and (grain size is high) and (acicularity is high) and(dislocation density is high) and (decomposition product ishigh) then (elastic modulus is medium).
7. If (volume fraction of β is medium) and (solution hard-ening is low) and (grain size is high) and (dislocationdensity is medium) then (elastic modulus is medium).
8. If (volume fraction of β is low) and (grain size is medium)and (acicularity is low) and (dislocation density is low) and(decomposition product is low) then (elastic modulus islow).
0.40.5
0.60.7
0.80.9
1 3040
5060
70
400
600
800
1000
1200
Grain size, micro m
Vol fraction of beta
Yie
ld s
treng
th, M
Pa
160170
180190
200210 250
300350
400450
500700
800
900
1000
1100
Hardness (precipitate), VHNHardness (solution), VHN
Yie
ld s
treng
th, M
Pa
810
1214
16
100
200
300
400400
600
800
1000
1200
Aspect ratio of grains
Dislocation density, 10 10/cm 2
Yie
ld s
treng
th, M
Pa
Fig. 11 – Surface views generated by the FIS describing the effects of (a) volume fraction of β and grain size, (b) solution andprecipitation hardening, and (c) dislocation density and aspect ratio of grains on yield strength.
40 50 60 70 80 90 100 110 120 13040
50
60
70
80
90
100
110
120
130
Measured Outputs
Pre
dict
ed O
utpu
ts
Fig. 12 – Scatter plot showing model predicted versusmeasured elastic modulus values.
j o u r n a l o f t h e m e c h a n i c a l b e h a v i o r o f b i o m e d i c a l m a t e r i a l s 5 3 ( 2 0 1 6 ) 3 5 0 – 3 6 5360
9. If (volume fraction of β is low) and (solution hardening ismedium) and (acicularity is medium) and (decomposition
product is low) then (elastic modulus is high).
10. If (volume fraction of β is low) and (solution hardening ismedium) and (grain size is high) and (acicularity ismedium) and (dislocation density is high) and (decom-position product is high) then (elastic modulus is low).
11. If (volume fraction of β is low) and (solution hardening islow) and (grain size is high) and (acicularity is low) and(dislocation density is high) and (decomposition product ishigh) then (elastic modulus is high-medium).
12. If (volume fraction of β is low) and (solution hardening islow) and (grain size is high) and (acicularity is medium)and (dislocation density is high) and (decompositionproduct is high) then (elastic modulus is high).
13. If (volume fraction of β is low) and (solution hardening islow) and (grain size is low) and (dislocation density ishigh) and (decomposition product is high) then (elasticmodulus is medium).
14. If (volume fraction of β is low) and (solution hardening islow) and (grain size is medium) and (dislocation density ismedium) then (elastic modulus is high).
15. If (volume fraction of β is low) and (solution hardening ishigh) and (grain size is medium) and (acicularity is medium)and (dislocation density is high) and (decomposition productis high) then (elastic modulus is low-medium).
j o u r n a l o f t h e m e c h a n i c a l b e h a v i o r o f b i o m e d i c a l m a t e r i a l s 5 3 ( 2 0 1 6 ) 3 5 0 – 3 6 5 361
In this case also it can be seen that the volume fraction of β isdirectly correlated to the elastic modulus in the expectedmanner. All other microstructural features have highly compli-cated relation with the output. The model predictions, as shownin Fig. 12, are found to be good enough for all practical purposes.
The surface views for input/output relationships gener-ated by the FIS from the elicited rules are shown in Fig. 13. Itcan be seen that the grain size, the solid solution hardeningand dislocation density do not have any significant effects onthe elastic modulus. The elastic modulus initially increaseswith increase in β content, but decreases sharply with furtherincrease. This indicates that the two phase structure has ahigher elastic modulus. The elastic modulus is lowest withmoderate precipitation hardening. As it can be seen in thecase of yield strength also, the role of β decompositionproducts are ‘tricky’ and may require further investigations.The increase in acicularity of the structure decreases theelastic modulus; this may indicate the formation of α′ and α′′.
4. Alloy design
To design an alloy for orthopedic and dental applications theprimary considerations are its mechanical properties and
0.40.5
0.60.7
0.80.9
1 2030
4050
6070
50
60
70
80
90
100
Grain size, m
icro m
Volume fraction of beta
Ela
stic
mod
ulus
, GP
a
810
1214
16 100
75
80
85
90
95
100
105
110
115
Aspect ratio
Ela
stic
mod
ulus
, GP
a
Fig. 13 – Surface views generated by the FIS describing the effecprecipitation hardening, and (c) aspect ratio of grains and disloc
biocompatibility. In the case of mechanical properties, on
the one hand the strength of the material should be high
enough to bear the load, and on the other hand, the elastic
modulus of the alloy has to be as close as possible to the
stiffness of the bone to avoid the problem stress shielding. As
these two objectives are conflicting in nature, multi-objective
optimization may be a solution for effective design. In this
case, evolutionary algorithm based RSSA is used to locate the
Pareto solutions for designing the alloys. As in the case of
multi-objective optimization a unique optimum solution may
not be found, instead of the several non-dominated solutions,
which are individually better in at least one objective com-
pared to other such solutions, for the Pareto front may be the
compromise solution. The developed rule-based models are
now used as the evaluation function of the two mechanical
properties (viz. yield strength and elastic modulus). As the
alloy will be used for the aged section of the masses, alloy
costs represent the third objective of the present problem. As
the cost of the processing is same for all Ti alloys, the cost of
the Ti and other alloying additions are only considered to
create the objective function. A simple weighted summation
method (amount of the alloying element multiplied by its
cost) is employed. The costs of the alloying elements are
160170
180190
200210 250
300350
400450
500
70
75
80
85
90
95
100
Hardness (precipitate), VHN
Hardness (solution), VHN
Ela
stic
mod
ulus
, GP
a
150 200 250 300 350
Dislocation density, 1010/cm2
ts of (a) volume fraction of β and grain size, (b) solution andation density on modulus of elasticity.
400600
8001000
12004646.5
4747.5
1400
1600
1800
2000
2200
2400
2600
2800
3000
Yield Strength (MPa)Elastic Modulus (GPa)
Cos
t (G
BP
/Kg)
Fig. 14 – Pareto solutions resulted from the multi-objectiveoptimization process.
Table 2 – Developed alloys from soft computingtechnique.
Alloy Al Zr Mo Nb Ta Sn Cr
1 6.9 4.1 7.0 0 0 9.9 10.12 2.1 3.9 10.1 4.9 5.1 8.0 5.83 2.0 0 0 0 0 12.8 18.2
Table 3 – Yield strength calculated from Vickershardness.
Alloy Processing Load (gf) Dwell time(s)
HV (GPa) YS (GPa)
1 AC 100 15 3.89 1.113WQ 100 15 3.8 1.082HR 100 15 3.753 1.065
2 AC 100 15 2.90 0.767WQ 100 15 2.77 0.722HR 100 15 2.902 0.767
3 AC 100 15 3.58 1.005WQ 100 15 3.78 1.075HR 100 15 3.69 1.044
Ti-6Al-4V
Wrought 100 15 3.1451 0.851
Table 4 – Elastic modulus of the alloys.
Alloy Processing Load (gf) Dwell time (s) HK (GPa) E (GPa)
1 AC 50 15 4.47 72.8WQ 50 15 4.6239 46.2HR 50 15 4.98 51.5
2 AC 50 15 3.3924 100.9WQ 50 15 3.832 41.7HR 50 15 3.558 80.0
3 AC 50 15 4.559 45.1WQ 50 15 4.691 52.2HR 50 15 4.64 72.8
Ti-6Al-4V
Wrought 50 15 3.86 88.4
j o u r n a l o f t h e m e c h a n i c a l b e h a v i o r o f b i o m e d i c a l m a t e r i a l s 5 3 ( 2 0 1 6 ) 3 5 0 – 3 6 5362
mostly collected from Ref. ⟨http://www.metalfirst.com⟩) and
are converted to GBP/kg. The objective function thus gener-
ated can be described as follows:
Cost = 1.5*Al + 315*V + 21*Zr + 24*Mo + 98*Nb + 270*Ta+ 0.5*Fe + 15*Sn + 8*Cr + 18*Ti (5)
∑[Ti = 100 − (all elements)]
The biocompatibility issue was taken into account using
two measures, viz. the coefficient of fibroblast outgrowth and
the relative growth rate of L929 cells. The published values for
the elements (Kuroda et al., 1998) are weighted according to
the percentage added and summed, as in Eqs. (6) and (7). The
relation is constrained to a value (Z100), which is described
as safe for orthopedic implants.Coefficient of fibroblast outgrowth
=0.8*Al + 0*V + 1.18*Zr + 1.18*Nb + 1.17*Ta + 0*Mo+ 1.15*Cr + 0.55*Fe + 1.3* Sn + 1.57*Ti (6)
Relative growth rate of L929 cell
=1.2*Al + 0*V + 1.1*Zr + 1.0*Nb + 0.9*Ta + 1.12*Mo+ 0.94*Cr + + 0*Fe + 1.3* Sn + 1.04*Ti (7)
The aluminum equivalent, suggested to be less than 9, for
maintaining the toughness of the alloy at adequate level, the
third constraint imposed. Ignoring the oxygen, nitrogen and
carbon content, it is described by Eq. (8) (Joshi; Weiss and
Semiatin., 1998):
Al Sn ZrAluminium Equivalent = + /3 + /6 (8)
The multi-objective optimization using the above three
objectives and three constraints is implemented using Multi-
objective Reduced Space Searching Algorithm (MO-RSSA)
(Zhang and Mahfouf, 2010). In the MO-RSSA algorithm,
several important parameters are used to control the optimi-
zation process, such as the decreasing parameter C1, the
increasing parameter C2, the changing ratio K, the decreasing
threshold m and the frequency parameter H. The decreasing
parameter C1 and the increasing parameter C2 are used to
choose the right moment for shifting the search space. The
scale of the search space variation is determined from the
changing ratio K. The decreasing threshold m controls the
achievable minimum size of the search space. The frequency
parameter H relates to the frequency of the weight changing
in the random weighted aggregation mechanism, which was
designed to solve the problems with multiple objectives. In
this work, the following parameters were used in MO-RSSA:
C1¼3, C2¼1, K¼0.5, m¼20, and H¼1000.The Pareto front developed through the multi-objective
optimization is shown in Fig. 14. The solutions have a
significant variation in the strength and cost of the alloys,
but the elastic modulus is within a narrow range. The detail
analysis of the non-dominated solutions is reported elsewhere
(Datta et al., 2013). The solutions at the knee region of the
Pareto front are considered for further studies, and based on
which three alloys are designed for experimantal validation.
j o u r n a l o f t h e m e c h a n i c a l b e h a v i o r o f b i o m e d i c a l m a t e r i a l s 5 3 ( 2 0 1 6 ) 3 5 0 – 3 6 5 363
5. Experimental validation
Designed Ti alloys are developed from under vaccum melting
furnace due to the high oxidation tendency of a few ele-
ments. The composition of the alloys is shown in the Table 2.
Three developed as cast Ti alloys are homoginized at 900 1C
and furnace cooled. Then again alloys are solutionized at 950,
700 and 850 1C for alloys 1, 2 and 3, respectively, with soaking
time of 1 h. The solutionizing of the alloys are done above
their β transus temperature, which are calculated from
literature (Guo et al., 2005). After homogenization, one set of
samples are water quench, and the other set are air cooled. A
portion of the alloys are hot rolled with a 20% deformation.The strength and elastic modulus of the developed alloys
are measured using standard micro-hardness techniques.
The samples are subjected to micro-vickers for measuring
the yield strength in Leco Micro hardness Tester LM 100. The
yield strength of the samples are calculated from the follow-
ing equation (Takakuwa et al., 2013).
σ H= (332 ± 18)· − (218 ± 30) (9)y V
where sy is the yield strength and HV is the Vickers hardness.The results are given in Table 3 along with the yield strengthof commercial wrought Ti-6Al-4V measured in the same way.The processing conditions AC, WQ and HR denote air cooled,
Fig. 15 – Light micrographs of (a) Alloy 1, (b) Alloy
water quenched and hot rolled respectively. It is seen that inthe case of Alloy 2, the strength level is lower than Ti-6Al-4V,the rest two alloys have superior strength levels. It shouldalso be noted here that the amount of β stabilizers ismaximum in Alloy 2. Hence, the presence of a higher amountof β may have reduced the strength.
The eastic modulus of the alloys are measured through
Knoop micro-hardness method. It is known that the Knoop
hardness
H PL
= = 14.229( ) (10)K 2
where P is the applied test load in N, L is the long indentationdiagonal lengths in m, and 14.229 is the geometrical constantof the diamond pyramid (Güdera et al., 2011). Again it isreported that w is the width of the indentation then
wL
wL
HE
= ′′
− 0.45 (11)K
where W′/L′ theoretical ratio of length and width and is equalto 0.1406, calculated from the geometrical characteristics ofthe Knoop indenter. Thus, the equation becomes as follows:
wL
HE
= 0.1406 − 0.45 (12)K
The elastic modulus E of the three different compositional
Ti alloys, as measured through Knoop hardness method in
2 and (c) Alloy 3 at water quenched condition.
j o u r n a l o f t h e m e c h a n i c a l b e h a v i o r o f b i o m e d i c a l m a t e r i a l s 5 3 ( 2 0 1 6 ) 3 5 0 – 3 6 5364
the same hardness testing machine as above, and theirrespective processing conditions are shown in the Table 4.It can be seen that though the E values varied extensivelywith processing conditions but most of the results areencouraging and almost up to expectations. In all cases, theE values are consistently low in water quenched conditions.Alloy 2, which did not provide a good strength, has the lowestE at water quenched condition. But the other alloys also hadcomparable results. A special mention should be made aboutAlloy 3; in this alloy the chief alloying addition is Cr to makethe alloy less expensive. Still it has provided a good combina-tion of high strength and low modulus of elasticity.
A priliminary study of the microstructure of the alloys inwater quenched conditions was carried out using lightmicroscope (Fig. 15). The alloys was etched using Kroll'sreagent, which is mostly suitable for Ti alloys (Jovanovic'et al., 2006). The micrographs show that in the cases of Alloys1 and 3 the second phase (presumably α) is present within theβ matrix significantly, where as in case of Alloy 2 the amountof second phase is significantly low. This explains the lowstrength and elastic modulus of Alloy 2. The microstructureof the three alloys is significantly different from each other,and thus requires in depth characterization. It is worth herethat the reason for the variations of the elastic modulus dueto different processing need to be explored. The presence ofother phases, such as α′, α′′ and ω, is known to have a stronginfluence on the modulus (Weiss and Semiatin., 1998; Ozakiet al., 2004; Hon et al., 2003). Thus, the analysis of themicrostructural constituents may provide clues as to thebehavior of the alloys at different processing conditions.
6. Conclusions
1. Prior knowledge in imprecise linguistic form could besuccessfully utilized for developing predictive models forthe mechanical properties of Ti alloys using fuzzy logic.
2. Prior experimental finding may be used simultaneously torefine and develop a semi-empirical two-layered fuzzyinference system, incorporating the microstructuralfeatures.
3. The fuzzy models for strength and elastic modulus may beused as objective functions for the multi-objective optimi-zation method using RSSA.
4. The Pareto solutions provide sufficient information fordesigning alloys. The solutions for the knee region of thePareto front are used for decision making.
5. The experimental validation of the designed alloy provideencouraging results in terms of strength and elastic mod-ulus of the alloys. Further in depth chracterization of thealloy is sure to establish a few novel alloy systems superiorin properties than the existing ones.
Acknowledgments
Prof. S. Datta and Prof. M. Mahfouf are grateful to RoyalAcademy of Engineering, UK for their support to the presentinvestigation in the Research Exchange scheme. N. Sultana is
grateful to Council of Scientific and Industrial Research, India.
All authors wish to thank all reviewers for their comments
which helped to improve the quality of this paper.
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