Chapter 7 Chapter 7 Conclusions and Future Scopes
3.4 Theoretical Modeling to Predict the Elastic Modulus of the Composites
0 5 10 15 20 25 30 35
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
Tensile strength (MPa)
Shear strength (MPa)
Shear
Tensile
Figure 3.11 Tensile strength vs. shear strength at the failure region
Chapter 3 Investigation and Evaluation of Mechanical Properties…
l cem
comp t cem
pp l cem t cem
1 2 l V
Y 3 d 5 1 2 V
Y 8 1 V 8 1 V
(3.4)
where, Ycomp and Ypp are the elastic modulus of composite and polypropylene respectively,
where
cem pp l
cem pp
Y 1
Y
Y 2( l / d ) Y
and
cem pp t
cem pp
Y 1
Y
Y 2
Y
where, Vcem denotes volume fraction of cement into polypropylene and l/d is the average aspect ratio of cement particles. In this work the cement particles were taken as (wt/wt) % and the volume fraction was calculated. Weight fraction of cement is expressed as:
cem cem
W w
100
where wcem is weight of the cement in polypropylene matrix. Similarly, the Volume fraction of cement is expressed as
comp cem cem
cem
V W
where ρcomp and ρcem are the theoretical specific gravity of composites and Portland Pozzolana Cement respectively. Further, according to modified series model, the composite elastic modulus can be written as (Cox, 1952; Jonathan et al., 2006):
Ycomp
l 0 cemY YppVcem Ypp (3.5) where, l denotes the length efficiency factor which can be written as2 p
l 2 p
Tanh a l
d e ( p 1 ) ( p 1 )
1 l p( e 1 )
a d
where, p a l d
; pp
cem cem
a 3Y
2Y ln(V )
; where, 0= 1/5 (Orientation efficiency factor for
random orientation) (Jonathan et al., 2006).
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Theoretical results are obtained using eqn. (3.4) and (3.5) and experimental results have been shown in the Figure 3.12. The experimental results make close agreement to the theoretical results when the aspect ratio (l/d) is considered as 45.
0 500 1000 1500 2000 2500
0 5 10 15
Elastic modulus (MPa)
Cement content (wt%)
Experimental results Halpin-Tsai model
Modified rule of mixing model
Figure 3.12 Comparison of experimental and theoretical values of Elastic modulus of cement- polypropylene composites
Both models have two factors in common, the stiffness is predicted in a scale taking into account both volume fraction and aspect ratio. In both the cases a more or less linear increase of modulus with volume fraction is predicted. This suggests that a good measure of the reinforcement is given by comp
filler
dY
dV at low volume fraction of filler (Vfiller). This takes into account for both the magnitude of the stiffness increase and the amount of fibre required to achieve it and has the advantage that it can easily be extracted from experimental data. The factor, comp
filler
dY
dV will be used to represent the magnitude of the reinforcement effect. For simplification, this quantity is written as comp
filler
dY
dV and refer to it as the reinforcement. This will be used to compare the results with the literature. Jonathan et al., (2006) have given the expression of the magnitude of the reinforcement as eqn.(3.6).
Chapter 3 Investigation and Evaluation of Mechanical Properties…
comp
l 0 filler mat filler
dY Y Y
dV (3.6)
From the eqn. (3.5), it is observed that l is the function of Vfiller, however past works, neglected the counter part of the derivatives of the main eqn. (3.5). Since lis the function of Vfiller the approximate derivatives of the eqn. (3.6) can be expressed as,
comp l
l 0 cem pp cem 0 cem
cem cem
dY Y Y V Y
dV V
Thus, on substitution the final expression is obtained as, after substituting
l 2 l
2a 2a
d d
2 2 2
0 cem
l l
2a 2a
d d
pp comp
cem pp
0 l cem
l e 1 e 1 l
a a a Y
d d
e 1 e 1
3Y l d
dY Y Y
dV
(3.7)
where, l/d is the average aspect ratio of the cement particles and pp
cem cem
a 3Y
2Y ln(V )
.
Similarly, for the Halpin and Kardos (Halpin and kardos, 1976) model, the Halpin-Tsai model for random orientation and the magnitude of the reinforcement is approximated as given below eqn. (3.8):
l
comp t
pp 2 pp 2
cem l cem t cem
1 2 l
dY 3 d 15
Y Y
dV 8 1 V 8 1 V
(3.8)
Elastic modulus is calculated using the modified Halpin-Tsai model with varying aspect ratio and represented in the Figure 3.13(a). It is anticipated that the elastic modulus varies with the aspect ratio of the filler materials according to the relationship given in eqn. (3.7). As shown in the Figure 3.13(a), the optimum aspect ratio of the filler materials is calculated as 213 when the modulus is almost saturated. Further, it is also observed that the modulus of the composite depends on the percentage of the filler materials as well and the modulus reinforcement depends on the aspect ratio (Figure 3.13(b)) as well. Modulus reinforcement,
comp cem
dY
dV is calculated using eqn. (3.7), with Ypp = 1325.286 MPa, Ycem = 20000 MPa for a volume fraction of 1.5667%, 3.2667% and 5.05% of cement. It is interesting to note that
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since the modulus reinforcement is a rate of change of elastic modulus with respect to the aspect ratio and do not affect by the filler volume fraction and can be justified from the Figure 3.13(b).
3.4.1 Results and Discussions
As shown in the Figure 3.13(c), the modulus reinforcement versus elastic modulus of all the composite samples show the slope of the straight lines of 5%, 10% and 15% cement filled composite are approximately 89º, 88º and 87º respectively. This signifies that for each 5%
cement loading enhancement, the slope decreases by 1º.
1200 1300 1400 1500 1600 1700 1800 1900
1 10 100 1000
Elastic modulus (MPa)
ln (l/d)
5%
10%
15%
(a)
Chapter 3 Investigation and Evaluation of Mechanical Properties…
3000 4000 5000 6000 7000 8000 9000 10000
1 10 100 1000
dYcomp/dVcem(MPa)
ln (l/d)
5%
10%
15%
(b)
3000 4000 5000 6000 7000 8000 9000 10000
1300 1400 1500 1600 1700 1800 1900
dYcomp/dVcem(MPa)
Elastic modulus (MPa)
5%
10%
15%
(c)
Figure 3.13 (a) Elastic modulus vs. aspect ratio of composite samples using modified Halpin-Tsai model, (b) Modulus reinforcement vs. aspect ratio plot and (c) Modulus reinforcement vs. elastic
modulus
Similarly, the elastic modulus is also calculated using the rule-of-mixing model (Jonathan et al., 2006) with varying aspect ratio and presented in the Figure 3.14(a). It is observed that the elastic modulus dramatically varies with the aspect ratio and stabilized when the aspect ratio reaches about 220. Further, from the result, it is also observed that the elastic modulus increases when the filler concentration increases. From the Figure 3.13(a), the optimum aspect ratio of the filler materials is also calculated and approximately 213 when the modulus
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is almost saturated. Here, an interesting point i.e, when the aspect ratio is less than 7 the elastic modulus of composite is less than the pure polymer which is insignificant for the composites. At the same time, when the aspect ratio is 7, all the composites (5%, 10% and 15% cement filled composite) are showing approximately the same elastic modulus and is equal to the elastic modulus of pure polypropylene. In addition, it is also observed that the modulus of the composite depends on the percentage of the filler materials as well. Similarly, the modulus-reinforcement is also depends on the aspect ratio (Figure 3.14(b)) where, modulus reinforcement is calculated using eqn. (3.7), on modification of the rule-of-mixing model.
1250 1300 1350 1400 1450 1500
1 10 100 1000
Elastic modulus (MPa)
ln (l/d)
5%
10%
15%
(a)
Chapter 3 Investigation and Evaluation of Mechanical Properties…
-1500 -1000 -500 0 500 1000 1500 2000 2500 3000
1 10 100 1000
dYcomp/dVcem(MPa)
ln (l/d)
5%
10%
15%
(b)
-1250 -250 750 1750 2750 3750
1250 1300 1350 1400 1450 1500
dYcomp/dVcem(MPa)
Elastic modulus (MPa)
5%
10%
15%
(c)
Figure 3.14 (a) Elastic modulus vs. aspect ratio of composite samples using rule of mixing, (b) Modulus reinforcement vs. aspect ratio and (c) Modulus reinforcement vs. elastic modulus
The approximate derivative of the eqn. (3.5) is expressed as in the eqn. (3.7). The calculation is being done with Epp = 1325.286 MPa, Ecem = 20000 MPa for a volume fraction of 1.5667%, 3.2667% and 5.05% of cement materials. It is worth to be noted that the modulus reinforcement or the reinforcement effect is a rate of change of elastic modulus with respect to the aspect ratio and thus remain unaffected by the filler volume-fraction, which is demonstrated in the Figure 3.13(b). While in the case of rule-of-mixing model, with an aspect
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ratio range 2-75, the reinforcement effect is observed to be affected by the volume fraction of the filler materials (Figure 3.14(b)).
Similar to the analysis carried out as shown in the Figure 3.13(c), the modulus reinforcement versus elastic modulus of all the composite samples show the linear relationship which is presented in the Figure 3.14(c). The slope of the straight lines of 5%, 10% and 15% cement filled composites are approximately 89º, 88º and 87º respectively. This signifies that for each 5% cement particles increment, the slope is decreased by 1º.