synthesis and characterization of aluminum matrix ......synthesis and characterization of aluminum...

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Synthesis and Characterization of Aluminum Matrix Composites Reinforced with Various Ratio of TiC for Light Devices MALEK ALI 2 , M. I. FADHEL 1 , M. A. ALGHOUL 2 , A. ZAHARIM 2 AND K. SOPIAN 2 1 Faculty of Engineering and Technology Multimedia University Jalan Ayer Keroh Lama 75450, Melaka 2 Solar Energy Research Institute Universiti Kebangsaan Malaysia 43600 UKM Bangi, Selangor MALAYSIA Email: [email protected] , [email protected] , [email protected] Abstract:- Powder metallurgy technique has been used to fabricate aluminum matrix composites reinforced by different weight ratios (Al-5, 15 and 25 wt. % TiC composites Al) to study the effect of different weight fractions on microstructure, mechanical, and electrical properties of the Al-TiC composites. The green compacts of Al-TiC composites were sintered for 3 hours at 500, 550, and 600 C. Hardness test, wear test, and electrical conductivity test were carried out on the Al-TiC composites. SEM techniques were used for microstructure examination of the Al-TiC composites. The presences of TiC particulates as reinforcement lead to a simultaneous increase in hardness and wear resistance of the composites. On the contrary, the electrical conductivity decreases marginally with increasing weight fraction of the TiC reinforcement in the aluminum matrix. Al-25wt. % TiC composites with 600C sintering temperature exhibited the best properties with microhardness value (63.7 Hv); wear rate (0.043 mm 3 /s). Al-5wt. % TiC composites with 600C sintering temperature exhibited the higher electrical conductivity value [0.285×107 (Ω-m)-1]. Keywords:- SEM techniques, microstructure examination, TiC composites 1 Introduction TiC reinforced metal matrix composites (MMCs) are increasingly being used in the automobile, aircraft, cutting tools, and space industries. Worldwide attention has been focused on the processing and fabrication of MMCs because of both favourable manufacturing costs and performance [1]. These composite materials also offer outstanding properties such as high strength- to-weight ratio, high torsion stiffness, good corrosion resistance and versatility to the designer. Two major processing techniques that have been found suitable for these composites are powder metallurgy and solidification processing [2]. As investigated by Yang et al (2003), hard TiC particles help improving the soft matrix in terms of hardness and wear resistance improvement that depends on the amount and uniformity of distribution of particles of TiC. The hardness is affected not only by the uniformity of distribution of particles in the matrix but also by the strength of the particle-matrix boundary and the mechanical properties of the matrix [3]. The Al-TiC composites occupy a unique position in the family of metal matrix composites due to their excellent wear/stiffness, strength-to- weight ratio with good mechanical properties. The most common applications of these composites are in commercial aerospace, space technology, automobile, general industrial and engineering structures. Typical examples are found in helicopter blade, automotive piston, engine block, cylinder liners, motorcycle brake disk, and valve engines [4,5]. Powder metallurgy technique is a promising technique to fabricate composites of Al reinforced with a hard ceramic phase. TiC particle-reinforced MMCs have been developed by many researchers because of the thermodynamically stability of TiC and the hardness and low density which it imparts to the composite [6, 7]. The present work Al-TiC composites have been fabricated and examined the effects of TiC ratio, and Sintering temperature on Microstructure, Mechanical Properties, and Electrical Conductivity in detail. It has been observed that Al-TiC has better strength than other Al based composites having same volume fraction of reinforced particles. Models and Methods in Applied Sciences ISBN: 978-1-61804-082-4 169

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Page 1: Synthesis and Characterization of Aluminum Matrix ......Synthesis and Characterization of Aluminum Matrix Composites Reinforced with Various Ratio of TiC for Light Devices MALEK ALI2,

Synthesis and Characterization of Aluminum Matrix Composites

Reinforced with Various Ratio of TiC for Light Devices

MALEK ALI2, M. I. FADHEL

1, M. A. ALGHOUL

2 , A. ZAHARIM

2 AND K. SOPIAN

2

1Faculty of Engineering and Technology

Multimedia University

Jalan Ayer Keroh Lama

75450, Melaka 2Solar Energy Research Institute

Universiti Kebangsaan Malaysia

43600 UKM Bangi, Selangor

MALAYSIA

Email: [email protected], [email protected], [email protected]

Abstract:- Powder metallurgy technique has been used to fabricate aluminum matrix composites reinforced by

different weight ratios (Al-5, 15 and 25 wt. % TiC composites Al) to study the effect of different weight

fractions on microstructure, mechanical, and electrical properties of the Al-TiC composites. The green

compacts of Al-TiC composites were sintered for 3 hours at 500, 550, and 600 ⁰C. Hardness test, wear test, and

electrical conductivity test were carried out on the Al-TiC composites. SEM techniques were used for

microstructure examination of the Al-TiC composites. The presences of TiC particulates as reinforcement lead

to a simultaneous increase in hardness and wear resistance of the composites. On the contrary, the electrical

conductivity decreases marginally with increasing weight fraction of the TiC reinforcement in the aluminum

matrix. Al-25wt. % TiC composites with 600⁰C sintering temperature exhibited the best properties with

microhardness value (63.7 Hv); wear rate (0.043 mm3/s). Al-5wt. % TiC composites with 600⁰C sintering

temperature exhibited the higher electrical conductivity value [0.285×107 (Ω-m)-1].

Keywords:- SEM techniques, microstructure examination, TiC composites

1 Introduction TiC reinforced metal matrix composites (MMCs)

are increasingly being used in the automobile,

aircraft, cutting tools, and space industries.

Worldwide attention has been focused on the

processing and fabrication of MMCs because of

both favourable manufacturing costs and

performance [1]. These composite materials also

offer outstanding properties such as high strength-

to-weight ratio, high torsion stiffness, good

corrosion resistance and versatility to the designer.

Two major processing techniques that have been

found suitable for these composites are powder

metallurgy and solidification processing [2]. As

investigated by Yang et al (2003), hard TiC particles

help improving the soft matrix in terms of hardness

and wear resistance improvement that depends on

the amount and uniformity of distribution of

particles of TiC. The hardness is affected not only

by the uniformity of distribution of particles in the

matrix but also by the strength of the particle-matrix

boundary and the mechanical properties of the

matrix [3]. The Al-TiC composites occupy a unique

position in the family of metal matrix composites

due to their excellent wear/stiffness, strength-to-

weight ratio with good mechanical properties. The

most common applications of these composites are

in commercial aerospace, space technology,

automobile, general industrial and engineering

structures. Typical examples are found in helicopter

blade, automotive piston, engine block, cylinder

liners, motorcycle brake disk, and valve engines

[4,5]. Powder metallurgy technique is a promising

technique to fabricate composites of Al reinforced

with a hard ceramic phase. TiC particle-reinforced

MMCs have been developed by many researchers

because of the thermodynamically stability of TiC

and the hardness and low density which it imparts to

the composite [6, 7].

The present work Al-TiC composites have been

fabricated and examined the effects of TiC ratio, and

Sintering temperature on Microstructure,

Mechanical Properties, and Electrical Conductivity

in detail. It has been observed that Al-TiC has better

strength than other Al based composites having

same volume fraction of reinforced particles.

Models and Methods in Applied Sciences

ISBN: 978-1-61804-082-4 169

Page 2: Synthesis and Characterization of Aluminum Matrix ......Synthesis and Characterization of Aluminum Matrix Composites Reinforced with Various Ratio of TiC for Light Devices MALEK ALI2,

2 Experimental The aluminum powder used was of purity 99.9%

and average particle size of less than 47.5 µm. TiC

particles were in the range 290.3-379.6 nm, mixing

of powders of Al and TiC with four different

compositions, i.e. 15%, 20%, 25%, 30% of TiC (by

weight) was prepared. Powder mixtures were

subjected to normal mixing for 2 hours at the speed

of 20 rpm. Mixtures of powders were manually

compacted into pellets by 750 MPa applied for 3

min. 0.5g of the powder mixture was pressed into 1

pellet in a steel mold of 10mm internal diameter.

Zinc stearate was used as a die wall lubricant. The

green compacts of Al-TiC composites were sintered

by tube furnace under protective argon gas

atmosphere for 3 hours at 500, 550, and 600 ⁰C;

SEM technique was used for microstructure

examination of the Al-TiC composites. The

hardness test of the sample is done by using micro

hardness Vickers testing machine. The load applied

on the sample is 0.5kgf, the micro-Vickers hardness

was measured at 5 points from the along the surface

of the samples. Dry sliding wear tests were

performed by pin-on-disc apparatus. Sliding

distance and velocity were kept constant at 282

meters and 150 rpm respectively for each specimen

with (320) grit size of abrasive paper and a constant

load. The Abrasive paper was changed after each

test. Each test was repeated three times and the

average value was taken.

3 Results and discussion

3.1 Effect of Sintering Temperature on

icrostructure Mechanical and physical properties of the

composites are highly dependent on its

microstructure. Microstructure of the composites

has been observed by using Scanning Electron

Microscopy (SEM). Figure 1 shows the SEM

micrographs of aluminum matrix composite

reinforced with 25wt. % of TiC particles at different

sintering temperatures. From Figure 1(a, b, and c), it

is observed that the sintered surfaces were porous

and the composite has small grains of TiC when it

sintered at 500 ºC. With the increase in sintering

temperature, the number of pores decreased and the

rate of grain growth apparently increased. Also the

microstructural study revealed that TiC clusters and

TiC-free regions were formed after consolidation at

the high sintering temperature (600ºC). It seems that

the aluminium matrix was locally moved during the

consolidation to fill the voids. They have suggested

that diffusion of the matrix into the interparticle

pores is responsible for this observation. Similarly, it

is suggestible that high diffusivity of the aluminium

matrix close to the melting point caused inter-

particle pore diffusion, causing the TiC clusters and

TiC-free areas [8].

(a) 500

oC

(b) 550

oC

(c) 600

oC

Fig. 1 SEM micrographs of the Al-25wt% TiC

composite at different sintering temperatures

Porosity

Models and Methods in Applied Sciences

ISBN: 978-1-61804-082-4 170

Page 3: Synthesis and Characterization of Aluminum Matrix ......Synthesis and Characterization of Aluminum Matrix Composites Reinforced with Various Ratio of TiC for Light Devices MALEK ALI2,

3.2 Effect of weight fraction of the TiC

particles on Microstructure The distribution of the reinforcement (TiC) particles

in the aluminium matrix has been observed by

scanning electron microscopy (SEM). Figure 2: a, b,

and c, shows the SEM micrograph of alumi

composites reinforced with 5, 15, and 25 wt% of

TiC respectively.

Fig. 2 SEM Micrograph of the composites after

600⁰C sintering temperature (a) Al-5%wt TiC

Al-15%wt TiC, and (c) Al-25%wt TiC.

It is clear from Figure 2 (a, b and c) that the

reinforcement particles of the composites are

TiC

(a)

(c)

TiC agglomeration

(b)

Porosity

Effect of weight fraction of the TiC

The distribution of the reinforcement (TiC) particles

in the aluminium matrix has been observed by

scanning electron microscopy (SEM). Figure 2: a, b,

and c, shows the SEM micrograph of aluminium

composites reinforced with 5, 15, and 25 wt% of

SEM Micrograph of the composites after

5%wt TiC, (b)

25%wt TiC.

It is clear from Figure 2 (a, b and c) that the

reinforcement particles of the composites are

embedded in the aluminium matrix. a small

agglomeration of TiC particles in the aluminium

matrix has been noticed and this is mainly due to

non homogeneity involved in mixing and blending

process carried out before sintering. The

microstructure evaluation also shows that, for a

given series of composites, the size of

particles in the composites increases as the TiC

content increases. This effect can be seen when

comparing the microstructures presented in Fig. 1

(a), (b) and (c). All composites were made using

fine TiC powder but the one with 25 wt. % of TiC

exhibits larger average TiC particles. The

microstructure of the TiC particles also reveals that

one big particle has been formed by the

agglomeration of many small particles together. The

small particles are distributed homogenously in

between the big particle and the particle size of the

small particles is uniform. However, some pores can

also be observed from Figure 2 and these are in

black colour. The formation of pores is mainly due

to the non uniformity of the initial powder particles.

3.3 Mechanical Properties

3.3.1 Hardness Results

The hardness measurement is known to be one of

the most informative and rapid methods to

determine the mechanical behaviour of

including composites. Yield and ultimate tensile

strengths, fatigue strength, wear resistance

often in good correlation

micro-Vickers hardness was measured at 3 points

from the surface of the samples. Hardness results of

the matrix and composites with 5, 15, and 25wt. %

of TiC are shown in Figure 3.

aluminium matrix. The relationship between the

wear rate of Al-TiC composites and the weight

fraction of TiC particulates is shown in Figure 4.

Fig. 3 Hardness of Al-TiC composites with different

wt% of TiC after sintering at 600

Al

TiC agglomeration

embedded in the aluminium matrix. a small

particles in the aluminium

matrix has been noticed and this is mainly due to

non homogeneity involved in mixing and blending

process carried out before sintering. The

microstructure evaluation also shows that, for a

given series of composites, the size of the TiC

particles in the composites increases as the TiC

content increases. This effect can be seen when

comparing the microstructures presented in Fig. 1

(a), (b) and (c). All composites were made using

fine TiC powder but the one with 25 wt. % of TiC

hibits larger average TiC particles. The

microstructure of the TiC particles also reveals that

one big particle has been formed by the

agglomeration of many small particles together. The

small particles are distributed homogenously in

cle and the particle size of the

small particles is uniform. However, some pores can

also be observed from Figure 2 and these are in

black colour. The formation of pores is mainly due

to the non uniformity of the initial powder particles.

perties

The hardness measurement is known to be one of

and rapid methods to

determine the mechanical behaviour of materials

including composites. Yield and ultimate tensile

fatigue strength, wear resistance, etc., are

with hardness [9]. The

Vickers hardness was measured at 3 points

from the surface of the samples. Hardness results of

the matrix and composites with 5, 15, and 25wt. %

of TiC are shown in Figure 3. resistance property of

aluminium matrix. The relationship between the

TiC composites and the weight

fraction of TiC particulates is shown in Figure 4.

TiC composites with different

wt% of TiC after sintering at 600 oC for 3 hours.

Models and Methods in Applied Sciences

ISBN: 978-1-61804-082-4 171

Page 4: Synthesis and Characterization of Aluminum Matrix ......Synthesis and Characterization of Aluminum Matrix Composites Reinforced with Various Ratio of TiC for Light Devices MALEK ALI2,

The composite shows higher hardness than the

base metal. The matrix of composites contains

higher dislocation density as a result of dislocations

generation due to large difference in thermal

expansion coefficient between the matrix and

reinforcement [10]. This leads to higher hardness in

the composites. Figure 3 shown that the hardness of

the composites are increasing with increasing the

weight fraction of the reinforcing TiC particulates in

aluminium matrix. This is due to the increase in the

hard phase (TiC) with a concomitant decrease in

ductility phase. However, a presence of TiC particle

refines the grain size. Finer grain size in composites

also leads to higher hardness. A higher hardness was

also associated with a lower porosity

3.3.2 Wear Results

It is well known that aluminium exhibit poor seizure

and wear resistance during sliding owing to their

softness. Such drawbacks can restrict their uses in

tribological environments. Generally, ceramic

reinforcing phases, confirm a beneficial effect by

improving the wear resistance property of

aluminium matrix. The relationship between the

wear rate of Al-TiC composites and the weight

fraction of TiC particulates is shown in Figure 4.

Fig. Error! No text of specified style in document.

Volumetric wear rates for Al-TiC composites with

different wt% of TiC after sintering 600

hours.

It is clear from Figure 4, that there is a significant

improvement in the wear resistance when the matrix

is reinforced with 5, 15, and 25wt.% TiC particles. A

considerable reduction in wear rate has been

observed for the Al-5wt.% TiC particulate

composite, compared to the unreinforced aluminum

due to the incorporation of TiC particles leading to

an increase in the hardness of composite also due to

better densification of the compacts and excellent

bonding between the matrix and TiC particles.

Moreover, the TiC particles provide improved wear

The composite shows higher hardness than the

base metal. The matrix of composites contains

higher dislocation density as a result of dislocations

generation due to large difference in thermal

expansion coefficient between the matrix and

This leads to higher hardness in

the composites. Figure 3 shown that the hardness of

the composites are increasing with increasing the

weight fraction of the reinforcing TiC particulates in

aluminium matrix. This is due to the increase in the

TiC) with a concomitant decrease in

ductility phase. However, a presence of TiC particle

refines the grain size. Finer grain size in composites

A higher hardness was

also associated with a lower porosity [11].

It is well known that aluminium exhibit poor seizure

and wear resistance during sliding owing to their

softness. Such drawbacks can restrict their uses in

tribological environments. Generally, ceramic

reinforcing phases, confirm a beneficial effect by

oving the wear resistance property of

aluminium matrix. The relationship between the

TiC composites and the weight

fraction of TiC particulates is shown in Figure 4.

Error! No text of specified style in document.

composites with

different wt% of TiC after sintering 600 oC for 3

It is clear from Figure 4, that there is a significant

improvement in the wear resistance when the matrix

is reinforced with 5, 15, and 25wt.% TiC particles. A

n in wear rate has been

5wt.% TiC particulate

composite, compared to the unreinforced aluminum

due to the incorporation of TiC particles leading to

an increase in the hardness of composite also due to

s and excellent

bonding between the matrix and TiC particles.

Moreover, the TiC particles provide improved wear

resistance to the composites than the unreinforced

by the formation of the mechanically mixed layer at

the interface of the composite pin and th

Further, there is a major difference between 5, 15,

and 25wt% TiC containing AMCs. It is evident from

the wear behavior that TiC particle reinforcement

has significant influence on the wear behavior of the

composites. The results show that the spec

rate of Al-TiC composites drastically decreases with

increasing TiC weight fraction. This is attributed to

high hardness of TiC particles, which resist the

penetration and microcutting action of abrasives

effectively, thus reducing the wear loss

order to investigate the wear mechanism, the

surfaces of the worn composites were examined

under SEM. Figures 5 (a, b and c) show typical

worn surfaces of the Al-5, 15, and 25 wt.% TiC

composites respectively. The worn surface of the

composites (Fig.5 (a, b and c)) appears small plough

groove and a little dimple. The abraded surfaces of

the composites also show scratches on the worn

surface, and there is little evidence of particulate

fracture, even in composites with the highest weight

fraction. The reason for this is explained as follows.

When the surface of composite initially comes in

contact with abrasive paper, adhesive contact

occurs. The TiC abrasive particles with sharp edges

then cause microploughing and grooving in the

surface of aluminium matrix. Therefore, materials in

the form of chips are removed from the grooves,

thereby exposing TiC particles. Therefore, the

increase in the weight fraction of TiC particles

appears to reduce the severity of grooving and

plastic deformation. This is

particulate phase is well bonded by Al

and strong enough to withstand abrasive wear

induced by the counter wear ring at applied load

[14].

3.4 Electrical Conductivity ResultsThe electrical conductivity was also measured on

the same samples to compare the electrical

conductivity of each composite material. The results

of electrical conductivity (σ) and electrical

resistivity (ρ) of Al-5wt.%, Al

TiC composites are given in Table 1. The results

show that the electrical resistivity increases with

increasing the weight fraction of TiC particles. The

electrical conductivity of the composites is shown in

Figure 6 as function of the weight fraction of TiC.

resistance to the composites than the unreinforced

by the formation of the mechanically mixed layer at

the interface of the composite pin and the disc.

Further, there is a major difference between 5, 15,

and 25wt% TiC containing AMCs. It is evident from

the wear behavior that TiC particle reinforcement

has significant influence on the wear behavior of the

composites. The results show that the specific wear

TiC composites drastically decreases with

increasing TiC weight fraction. This is attributed to

high hardness of TiC particles, which resist the

penetration and microcutting action of abrasives

effectively, thus reducing the wear loss [12,13]. In

order to investigate the wear mechanism, the

surfaces of the worn composites were examined

under SEM. Figures 5 (a, b and c) show typical

5, 15, and 25 wt.% TiC

composites respectively. The worn surface of the

(Fig.5 (a, b and c)) appears small plough

groove and a little dimple. The abraded surfaces of

the composites also show scratches on the worn

surface, and there is little evidence of particulate

fracture, even in composites with the highest weight

. The reason for this is explained as follows.

When the surface of composite initially comes in

contact with abrasive paper, adhesive contact

occurs. The TiC abrasive particles with sharp edges

then cause microploughing and grooving in the

nium matrix. Therefore, materials in

the form of chips are removed from the grooves,

thereby exposing TiC particles. Therefore, the

increase in the weight fraction of TiC particles

appears to reduce the severity of grooving and

plastic deformation. This is because the hard

particulate phase is well bonded by Al-based matrix

and strong enough to withstand abrasive wear

induced by the counter wear ring at applied load

Electrical Conductivity Results The electrical conductivity was also measured on

the same samples to compare the electrical

conductivity of each composite material. The results

of electrical conductivity (σ) and electrical

5wt.%, Al-15wt.%, Al-25wt.%

are given in Table 1. The results

show that the electrical resistivity increases with

increasing the weight fraction of TiC particles. The

electrical conductivity of the composites is shown in

Figure 6 as function of the weight fraction of TiC.

Models and Methods in Applied Sciences

ISBN: 978-1-61804-082-4 172

Page 5: Synthesis and Characterization of Aluminum Matrix ......Synthesis and Characterization of Aluminum Matrix Composites Reinforced with Various Ratio of TiC for Light Devices MALEK ALI2,

Fig. 5 Micrographs show worn surface of Al

(a) Al-5wt% TiC, (b) Al-15wt% TiC, (c) Al

TiC.

(a)

(b)

(c)

5 Micrographs show worn surface of Al-TiC,

15wt% TiC, (c) Al-25wt%

Table 1 Electrical conductivity of Al

25%wt TiC composites after sintering.

Composite Electrical

Resistivity

(ohm

Al pure 0.025

Al-5wt.% TiC 0.035

Al-15wt.% TiC 0.138

Al-25wt.% TiC 0.334

Fig. 6 Electrical conductivity for Al

with different wt.% of TiC.

4 Conclusions The Al-TiC composite has been synthesis

successfully by different weight ratios (Al

25 wt. % TiC composites Al) and different sintering

Temperature (500, 550, and 600 ºC). With the

increase in sintering temperature, the number of

pores decreased and the rate of grain growth

apparently increased. Different weight ratios of TiC

as reinforcement affect on microstructure,

mechanical, and electrical properties of the Al

composites. Al-25wt. % TiC composites with 600 ºC

sintering temperature exhibit

with microhardness value (63.7 Hv); wear rate

(0.043 mm3/s) as increasing of TiC ratio lead to

increase mechanical properties, Whereas Matrix. Al

5wt. % TiC composites with 600 ºC sintering

temperature exhibited the higher electrical

conductivity value [0.285×107 (

Electrical conductivity of Al-5, 15, and

25%wt TiC composites after sintering.

Electrical

Resistivity

(ohm-cm)

Electrical

Conductivity

(Ω-m)-1

0.025 0.384×107

0.035 0.285×107

0.138 0.072×107

0.334 0.029×107

Electrical conductivity for Al-TiC composites

with different wt.% of TiC.

TiC composite has been synthesis

successfully by different weight ratios (Al-5, 15 and

25 wt. % TiC composites Al) and different sintering

Temperature (500, 550, and 600 ºC). With the

increase in sintering temperature, the number of

and the rate of grain growth

apparently increased. Different weight ratios of TiC

as reinforcement affect on microstructure,

mechanical, and electrical properties of the Al-TiC

25wt. % TiC composites with 600 ºC

sintering temperature exhibited the best properties

with microhardness value (63.7 Hv); wear rate

/s) as increasing of TiC ratio lead to

increase mechanical properties, Whereas Matrix. Al-

5wt. % TiC composites with 600 ºC sintering

temperature exhibited the higher electrical

conductivity value [0.285×107 (Ω-m)-1].

Models and Methods in Applied Sciences

ISBN: 978-1-61804-082-4 173

Page 6: Synthesis and Characterization of Aluminum Matrix ......Synthesis and Characterization of Aluminum Matrix Composites Reinforced with Various Ratio of TiC for Light Devices MALEK ALI2,

References:

[1] Ramesh, K. C. & Sagar, R. Fabrication of metal

matrix composite automotive parts, J. Adv.

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[2] Yuan, Q., Zheng, Y. & Yu, H. Mechanism of

synthesizing nanocrystalline TiC in different

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Hard Materials, 27, 2009, p. 696–700.

[3] Yang, Q., Senda, T., Ohmori A. Effect of

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[4] Nukami T. The growth of TiC particles in an

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[5] A. E. Karantzalis, A. Lekatou, , E. Georgatis,

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[6] Humphreys FJ, Basu A, Djazeb MR. The

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[7] Kennedy, A. R. & Wyatt, S. M. Characterising

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Models and Methods in Applied Sciences

ISBN: 978-1-61804-082-4 174