1-s2.0-S1751616115003239-main epr25.pdf

Available online at www.sciencedirect.com

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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,

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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 5 351

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


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


(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).


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.


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.


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.


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.


(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).


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.


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).


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


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.


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.


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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