International Conference on Mechanical, Production and Materials Engineering (ICMPME'2012) June 16-17, 2012, Bangkok
Thermal Expansion Coefficient for LeadGraphite and Lead-Iron Metal Matrix
Composites
Jagannath K1, S.S.Sharma2, Chandrashekar Bhat3, Raghavendra Prabhu4
Abstract--One of the significant features of a composite material
is tailorability of its material properties. Coefficient of thermal
expansion (CTE) of a composite material is known to play a key role
in its application area. It has been realized that a state of micro stress
often exists between the phase of the matrix and reinforcement.
Difference in thermal expansion of the individual phases produces
stress which indirectly affects the strength properties and modes of
failure. In the present study, coefficient of thermal expansion is
measured using metroscope, with a least count of 0.2 microns.
Nichrome wire embedded specimens are used for the experiment.
The main findings from the experiment is that the coefficient of
thermal expansion for both lead-graphite and lead-iron composites
increase with the increase in temperature. The rate of coefficient of
thermal expansion decreases with increase in weight percentage of
graphite or iron.
composite.The differences between these two models are
related to the different properties in this third phase. Siderids
et al. [5] derived a three phase model of the generalized selfconsistent model, which is an improved version of Kerner
model [6]. The models described above assume that the
interfaces are perfect mathematical surfaces and that the
adhesion between matrices is also perfect. However, in reality
a complex interaction develops around an inclusion. Areas of
imperfect bonding, permanent stresses due to shrinkage of the
composite phase during the curing period, high stress
gradients, stress singularities due to the rough surfaces of the
inclusion, voids microcracks, etc. mean that the perfect
conditions assumed don’t represent the reality.
The
interaction of the matrix with the surface of the reinforcement
inclusion during curing, restricts the free segmental and
molecular mobility of the matrix and thus creates a
constrained layer with different mechanical and physical
properties of the composite.
The thermal expansion
coefficient of a particle reinforced composite material for
lead- graphite and lead-iron composites studied and presented
in this paper.
Keywords--Metal matrix composite, Coefficient of thermal
expansion, Residual stress, Metroscope.
I. INTRODUCTION
M
ANY theoretical analyses, which define the thermal
properties of composites and provide equations for
predicting the thermal expansion coefficient, have been
reported in the literature [1,2]. Some of the derived equations
have been verified experimentally for some practical systems
but for other systems poor agreement was found between the
theory and the experimental results. The influence of thermal
expansion behaviour of MMC was demonstrated
experimentally [2] that thermal expansion coefficient did not
obey the law of mixture. Hasin in 1962 [3] assumed that a
composite is a collection of small volume elements of various
sizes and shapes, which densely filled in the bulk composites.
The paper considered the particulates as conglomerations of
spherical inclusions and shells, with the properties of the
matrix, surrounding the inclusions. In each volume element,
the volume fraction of the inclusion was equal to the total
content of the dispersed phase in the composite. Vander Poel
[4] considered a representative volume element (RVE)
consisting of the filler and the shell of matrix surrounded by a
third substance with properties such that the average state of
stress and strain was closer to the real situation in the
II METHODOLOGY
Coefficient of thermal expansion (CTE) is measured using
metroscope, which has a least count of 0.2 microns. Fig.1
illustrates the metroscope used for the experiment. Nichrome
wire embedded specimens are used for the experiment. Initial
reading is measured and the current is passed through the
conductor and temperature of the wire and outside
temperature of the specimen are measured. Final expansion
reading is taken using metroscope and difference in expansion
∆L is computed. Thermal expansion is computed using the
equation α=∆L/L∆T where L is the original length of the
sample and ∆L is the change in length over a temperature
interval ∆T.
Dr Jagannath1, Dr S.S.Sharma2 and Dr Chandrashekar Bhat3 are working
with MIT Manipal, India as Professors in the department of Mechanical and
Manufacturing Engg. (e-mail: [email protected])
Mr Raghavendra Prabhu4 is working with MIT as Assistant Professor,
Senior scale in the department of Mechanical and Manufacturing Engg
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International Conference on Mechanical, Production and Materials Engineering (ICMPME'2012) June 16-17, 2012, Bangkok
The coefficient of thermal expansion of a metal matrix
composite is not easy to predict precisely because it is
influenced by several factors such as matrix plasticity, size
and shape of the reinforcement, type of reinforcement,
distribution of the reinforcement voids in the metal matrix
composite etc.
The theoretical analysis for the determination of the thermal
expansion coefficient is based on the following assumptions:
(i) The particles and the matrix are elastic, isotropic and
homogeneous.
(ii) The particles are perfectly spherical in shape.
(iii)The particles are large in number, and their distribution
is uniform so that the composite may be regarded as a
quasi-homogeneous isotropic material.
(iv)The deformations applied to the composite are
small enough to
maintain linearity of stress–strain
relationships.
Fig.1 Universal Horizontal Metroscope
1. Setscrew
2. Hand wheel
3. Foot screw
4. Holder plate
5. Spiral microscope
6. Measuring element
7. Measuring sleeve
8. Measuring bolt
9. Adjustable object table
10. Measuring bolt
11. Tube sleeve
12. Setting knob
13. Sleeve slide
14. Clamp screw
15. Clamp handle
The expression for the thermal expansion coefficient can be
obtained by applying the classical theory of elasticity to the
representative volume element (RVE) whose mechanical
properties equal to the average properties of the composite.
Several models have been proposed for the prediction of
coefficient of thermal expansion of metal matrix composites
[2]. Among them, the seven simplest and the most commonly
used models are discussed in this paper. Tables I and II
illustrate the calculated value of CTE from different
theoretical models given in the literature [2] for the leadgraphite and lead-iron composites. The results clearly show
the reduction in coefficient of thermal expansion as the weight
percentage of reinforcement increases.
Specimens are prepared for measurement of thermal
expansion from the cast ingot using a micro-cut machine.
Sample specimens were polished using polishing machine
with 1 micron size diamond paste. Four samples of each
composite were tested under the same condition to verify the
reproducibility. The coefficients of thermal expansion were
measured in the temperature range 35 to 85 0C.
TABLE I. CTE FOR LEAD-GRAPHITE COMPOSITES FROM DIFFERENT THEORETICAL MODELS
Sl.
no
Models
1%
2%
3%
4%
5%
1
Mixture law
28.959
28.819
28.077
27.736
27.395
2
Turner equation
28.591
28.108
27.649
27.387
26.797
3
Kerner equation
28.971
28.843
28.214
27.584
27.155
4
Blackburn equation
28.896
28.694
28.494
28.196
27.81
5
Wang and Kwei equation
29.097
29.094
29.075
28.759
28.430
6
Tummala equation
28.964
28.828
28.692
28.556
28.42
7
Fahmi equation
29.053
28.806
28.359
28.013
27.766
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International Conference on Mechanical, Production and Materials Engineering (ICMPME'2012) June 16-17, 2012, Bangkok
TABLE II CTE FOR LEAD-IRON COMPOSITES FROM DIFFERENT THEORETICAL MODELS
Sl.
no
1
Models
1%
2%
3%
4%
5%
10%
15%
Mixture law
28.86 3
28.626
28.389
28.152
27.915
25.66
22.98
2
Turner equation
28.548
28.012
27.489
26.980
26.484
24.21
21.36
3
Kerner equation
28.851
28.603
28.356
28.108
27.861
26.15
24.67
4
Blackburn equation
28.957
28.813
28.669
28.523
28.376
27.18
25.34
5
Wang and Kwei equation
29.098
29.093
29.084
29.071
29.055
27.53
26.88
6
Tummala equation
28.946
28.793
28.640
28.487
28.334
27.24
25.46
7
Fahmi equation
29.008
28.916
28.822
28.727
28.631
27.54
24.97
super cooling is important for tribological applications of the
composites. Higher the degree of superheating may lead to
severe internal stresses in the composite which may lead to
cracking.
III RESULTS AND DISCUSSION
Tables III and IV give the coefficient of thermal expansion
obtained experimentally for graphite or iron reinforced lead
metal matrix composites. Same experimental data is presented
graphically in Figs.2 and 3. In both the composites, the
coefficient of thermal expansion decreases as the weight
percentage of particulates increases. Almost similar
coefficient of thermal expansion values are observed in the
composite, which indicate that higher particle density
decreases the thermal expansion.
Coeff. of thermal expansion, x10-6
30.0
For the matrix containing particulates as several phases,
upon cooling from processing temperature, residual stress
would have developed due to the difference in thermal
expansion and elastic properties of the matrix and particulates.
Thermal expansion of lead is greater than graphite where as
thermal expansion of iron is greater than graphite. A
quantitative analysis of residual stress resulting from the
thermal mismatch in particulate composites was determined
from the Turner model [2] which is expressed as:
d=
σ m (α mα f V f E f Em ∆T (Vm Em + V f E f ) -----(1)
29.5
29.0
28.5
28.0
27.5
27.0
26.5
26.0
0
1
2
3
4
5
6
Percentage of graphite
Fig.2 CTE for Lead-Graphite Composites
30
Coeff. of thermal expansion, x10-6
where σ m is the axial stress in the matrix material, the
subscripts m and f refer to matrix and particulates
respectively, E is the Young’s modulus, V is the volume
fraction, ∆T is the degree of super cooling and α is the
coefficient of thermal expansion. The expression (1) is used
to evaluate the thermal stresses in the matrix with a change in
temperature which is shown in Figs. 4 to 7. Figs 4 and 5 show
the internal stress developed during super cooling from the
molten state to room temperature for lead-graphite and leadiron composites. It is observed that stresses are linearly
decreasing as the degree of super cooling decreases. However,
the values of internal stresses for lead-iron composites are
more than the lead-graphite composites, since iron has more
thermal expansion than lead. Figs. 6 and 7 show the increase
in internal stresses of the composites as the degree of
superheating increases from 30 to 800C. The stress due to
28
26
24
22
0
2
4
6
8
10
12
Percentage of iron
Fig.3 CTE for Lead-Iron Composites
22
14
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International Conference on Mechanical, Production and Materials Engineering (ICMPME'2012) June 16-17, 2012, Bangkok
15000
14000
Graphite-1%
Graphite-2%
Graphite-3%
Graphite-4%
Graphite-5%
50000
Residual Stress (N/mm2)
Residual stress (N/mm2)
60000
40000
30000
20000
Graphite-1%
Graphite-2%
Graphite-3%
Graphite-4%
Graphite-5%
13000
12000
11000
10000
9000
8000
7000
6000
5000
4000
3000
2000
10000
1000
30
0
350
300
250
200
150
100
40
50
60
70
80
o
Degree of superheating ( C)
50
O
Degree of supercooling ( C)
Fig.6 Variations of Internal Stresses on the Lead-Graphite
composites with degree of superheating
Fig.4 Variations of Internal Stresses on the Lead-Graphite
composites with degree of supercooling
1600000
350000
1200000
1000000
Iron-1%
Iron-3%
Iron-5%
Iron-10%
Iron-15%
300000
Iron-1%
Iron-3%
Iron-5%
Iron-10%
Iron-15%
Residual stresses (N/mm2)
Residual stress (N/mm2)
1400000
800000
600000
400000
250000
200000
150000
100000
50000
200000
0
30
40
50
60
70
80
o
0
Degree of superheating ( C)
350
300
250
200
150
100
50
o
Degree of supercooling( C)
Fig.7 Variations of Internal Stresses on the Lead-Iron composites
with degree of superheating
Fig.5 Variations of Internal Stresses on the Lead-Iron composites
with degree of supercooling
TABLE III CTE FOR LEAD-GRAPHITE COMPOSITES
Wt.% of
Initial reading
Final reading
Difference
Initial
Graphite added
(mm)
(mm)
(mm)
1
73.3966
73.4689
0.0723
35
70
35
28.1445
2
73.4700
73.5613
0.0913
35
80
45
27.6152
temp.
Final
temp.
Change
CTE
in
X 10-6
temp.
3
73.9364
74.0269
0.0905
35
80
45
27.2075
4
73.9104
73.9898
0.0794
35
75
40
26.8568
5
73.4580
73.5259
0.0679
35
70
35
26.4096
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International Conference on Mechanical, Production and Materials Engineering (ICMPME'2012) June 16-17, 2012, Bangkok
TABLE IV CTE FOR LEAD-IRON COMPOSITES
Wt. % of
iron
Initial reading
Final reading
(mm)
(mm)
Difference
Initial temp.
Final temp.
(mm)
Change
CTE
in
X 10-6
temp.
added
1
72.8634
72.9580
0.0946
35
80
45
28.8515
2
72.9754
73.0565
0.0811
35
75
40
27.7833
3
74.1082
74.2005
0.0923
35
80
45
27.6772
4
73.5352
73.6346
0.0994
35
85
50
27.0346
5
74.3040
74.3831
0.0791
35
75
40
26.6136
IV CONCLUSIONS
The thermal expansion coefficient of metal matrix
composites reinforced with graphite and iron particulates in
lead is in the elastic region for the temperature range from 0
to 80 0 C.
The coefficient of thermal expansion for both lead-graphite
and lead-iron composites increase with increase in
temperature.
The rate of coefficient of thermal expansion decreases with
increase in weight percentage of graphite or iron.
REFERENCES
[1] Smagorinski M.E. and P.G. Tasant Rizor, Development of light composite
material with low coefficient of thermal expansion, Journal of Material
Science and Technology, Vol. 16, 2000, pp 883-862.
[2] Sharma S.C., Effect of ablate particles on the coefficient of thermal
expansion behaviour of the Al 6061 alloy composites, Journal of
Metallurgical and Materials Transactions A,Vol.31, 2000, pp. 773-779.
[3] Hasin Z., Thermal expansion in metal matrix composites, Journal of
Applied Mechanics, Vol.29. 1962, pp. 143-148.
[4] Vander Poel , Modern materials, Journal of Rheokacta, Vol.1, 1958, pp.
198-206.
[5] Siderids E., V.N. Kytoaulas and E. Kyrazi, Determination of thermal
expansion coefficient of particulate composites by the use of triphase model, Journal of Composites Science and Technology, Vol.2,
2000, pp.1-11
[6] Kerner E.H., The elastic and thermoelastic properties of composite media,
Proceedings of Physics Society, 1956, 69B, pp.808.
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