A numerical study on the thermal expansion coefficients of fiber

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where
θ
1
K
( K
c
− K
f
)
( K − K )
K
( K
m
− K
c
)
( K − K )
m
f
= , θ
2
=
(3.13)
K
c m f
K
c m f
In these equations K c , K f and K m represent the bulk moduli of the composite, filler
and matrix respectively. The simplicity of this approach is attractive, but it converts
the problem of calculating α c to knowledge of K c or an ability to calculate it from the
properties and volume fractions of the individual components.
3.4.1.5 Equation of Schapery
Schapery (1968) has derived expressions for the longitudinal and transverse
effective thermal expansion coefficients both for isotropic and anisotropic
composites consisting of isotropic phases, by employing extremum principles of
thermoelasticity.
He considered a specimen under a space wise uniform temperature in the form of
a rectangular parallelepiped whose edges are parallel to coordinate axes and with unit
volume. This specimen is statistically homogeneous and composed of n phases
(constituents), each of which has homogeneous mechanical and thermal properties
that are different from any other phase. No restriction is placed on the temperature
dependence of constituent properties. Maximum dimensions defining the specimen’s
structural inhomogeneity are assumed small compared to specimen dimensions and
the interaction between phases is considered to be purely mechanical and linear. Also
specimen and phases are assumed unstressed and unstrained when surface tractions
are zero and the temperature is at some reference value.
As a result, for a unidirectional two phase composite, the thermal expansion
coefficient in the longitudinal direction is
α
E
α
υ
+ E
α
υ
f f f m m m
1
= (3.14)
E
f
υf
+ E
mυm