A numerical study on the thermal expansion coefficients of fiber
A numerical study on the thermal expansion coefficients of fiber A numerical study on the thermal expansion coefficients of fiber
43 Square packing Figure 3.5 Idealized fiber packing arrangements. Hexagonal packing 3.4.1.7 Equation of Rosen and Hashin Rosen and Hashin (1970) derived expressions for the effective thermal expansion coefficients of multiphase composites and the summary of their approach for unidirectional fiber reinforced composites with transversely isotropic constituents is given as α α 1 2 = αˆ 1 = αˆ 2 + + + + + ( S11 − Ŝ11 ) ( α f1 − α m1) P11 + ( α f2 − α m2 ) 2P12 ( S − Ŝ ) ( α − α ) 2P + ( α − α ) 2( 12 12 [ ] [ P + P )] f1 m1 ( S12 − Ŝ12 ) ( α f1 − α m1) P11 + ( α f2 − α m2 ) ( S22 − Ŝ22 ) ( α f1 − α m1) P12 + ( α f2 − α m2 )( ( S − Ŝ ) ( α − α ) P + ( α − α )( 23 23 12 [ 2P12 ] [ P22 + P23 )] [ P + P )] f1 m1 12 f2 f2 m2 m2 23 22 33 23 (3.18) (3.19) where S S 11 12 1 = , S E = S 1 13 22 = S 1 33 1 = E −ν12 −ν = , S23 = E E 2 23 2 (3.20) and P ab (a,b=1,2,3) are
44 P P 11 33 2 A 22 − A = Det A A = P = 22 A12A 23 − A12A , P13 = P12 = Det A 2 2 A 22 − A12 A12 − A11A , P23 = Det A Det A 2 23 11 22 23 (3.21) where A ab is defined as (S fab - S mab ) and Det A is given by 2 2 ( A − A ) + 2A ( A A − A ) Det A = A (3.22) 11 22 23 12 12 23 12A 22 In above equations subscripts f and m refer to fiber and matrix and terms with and without a hat refer to volume average and effective composite properties respectively. S ab can be written for the effective property, the volume average property and the constituent properties. Composite volume average properties are obtained with the expression pˆ = p υ + p υ (3.23) f f m m and composite effective mechanical properties are given by Hashin (1979). It is very difficult to find the coefficients of thermal expansion of a composite using equations of Rosen and Hashin because, to determine composite effective mechanical properties the bulk modulus of the constituents and the composite should be determined. 3.4.1.8 Equation of Schneider Another consideration was made by Schneider (1971), who assumed a hexagonal arrangement of cylindrical fiber-matrix elements consisting of a fiber surrounded by a cylindrical matrix jacket. The equation for the longitudinal thermal expansion coefficient is identical to Schapery’s formula. For the transverse direction he derived the following equation
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- Page 25 and 26: 16 2.2.2.2 Carbon Fibers Carbon is
- Page 27 and 28: 18 use is in aircraft industry foll
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- Page 31 and 32: 22 1. Processing the conventional f
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- Page 77 and 78: 68 Longitudinal CTE (1/°C) 2.00E-0
- Page 79 and 80: 70 Longitudinal CTE (1/°C) 4.00E-0
- Page 81 and 82: 72 Longitudinal CTE (1/°C) 1.00E-0
- Page 83 and 84: 74 Ishikava, T., Koyama, K., & Koba
43<br />
Square packing<br />
Figure 3.5 Idealized <strong>fiber</strong> packing arrangements.<br />
Hexag<strong>on</strong>al packing<br />
3.4.1.7 Equati<strong>on</strong> <strong>of</strong> Rosen and Hashin<br />
Rosen and Hashin (1970) derived expressi<strong>on</strong>s for <strong>the</strong> effective <strong>the</strong>rmal expansi<strong>on</strong><br />
<strong>coefficients</strong> <strong>of</strong> multiphase composites and <strong>the</strong> summary <strong>of</strong> <strong>the</strong>ir approach for<br />
unidirecti<strong>on</strong>al <strong>fiber</strong> reinforced composites with transversely isotropic c<strong>on</strong>stituents is<br />
given as<br />
α<br />
α<br />
1<br />
2<br />
= αˆ<br />
1<br />
= αˆ<br />
2<br />
+<br />
+<br />
+<br />
+<br />
+<br />
( S11<br />
− Ŝ11<br />
) ( α<br />
f1<br />
− α<br />
m1) P11<br />
+ ( α<br />
f2<br />
− α<br />
m2<br />
) 2P12<br />
( S − Ŝ ) ( α − α ) 2P + ( α − α ) 2(<br />
12<br />
12<br />
[ ]<br />
[ P + P )]<br />
f1<br />
m1<br />
( S12<br />
− Ŝ12<br />
) ( α<br />
f1<br />
− α<br />
m1) P11<br />
+ ( α<br />
f2<br />
− α<br />
m2<br />
)<br />
( S22<br />
− Ŝ22<br />
) ( α<br />
f1<br />
− α<br />
m1) P12<br />
+ ( α<br />
f2<br />
− α<br />
m2<br />
)(<br />
( S − Ŝ ) ( α − α ) P + ( α − α )(<br />
23<br />
23<br />
12<br />
[ 2P12<br />
]<br />
[ P22<br />
+ P23<br />
)]<br />
[ P + P )]<br />
f1<br />
m1<br />
12<br />
f2<br />
f2<br />
m2<br />
m2<br />
23<br />
22<br />
33<br />
23<br />
(3.18)<br />
(3.19)<br />
where<br />
S<br />
S<br />
11<br />
12<br />
1<br />
= , S<br />
E<br />
= S<br />
1<br />
13<br />
22<br />
= S<br />
1<br />
33<br />
1<br />
=<br />
E<br />
−ν12<br />
−ν<br />
= , S23<br />
=<br />
E E<br />
2<br />
23<br />
2<br />
(3.20)<br />
and P ab (a,b=1,2,3) are
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