Composite Materials Research Progress

16
Jacquemin Frédéric and Fréour Sylvain
m I I i I I −1
I I
( L + L : R ) : ( L + L : R ) : L
I
m ΔC m−1
β = L :
: β (21)
m m
v ΔC
i=
r, m
The pseudomacroscopic coefficients of thermal expansion of the matrix can be deduced
from the inversion of the homogenization form (3) as follows:
( ) ( ) ( ) ⎥ ⎥
⎡
⎤
m I I
−
−
+ ⎢ i I I 1 I
I r r I I 1 r
L L : R : L + L : R : L : M − v L + L : R : L M
m m−1
r (22)
M = L :
:
⎢
⎣
i=
r, m
⎦
Form (22) can be easily rewritten for expressing the coefficients of thermal expansion of
the reinforcements, using the same replacement rules over the superscripts/subscripts:
m → r, r → m , than for the previous cases.
3.2.2. Application of Mori-Tanaka Estimates to the Identification of the Pseudomacroscopic
Properties of one Constituent Embedded in a Two-Constituents
Composite Material
3.2.2.1. Inverse Mori-Tanaka Elastic Model
In the present work, it is be considered, that the reinforcements are surrounded by the matrix,
thus, T m =I and (11) develops as follows:
I
( ) ( ) 1
m m r r r m r r
−
v L + v L : T : v I + v : T
L =
(23)
Thus, from (11) two alternate equations are obtained for identifying the pseudomacroscopic
stiffness of the composite ply constituents:
• On the first hand, the elastic properties of the matrix satisfies
I r r
( L - L ) : T
m
m I 1-
v
L = L +
(25)
m
v
Equation (25) is an implicit equation since both its left and right hand sides involve the
researched stiffness tensor L m .
• whereas, on the second hand, the elastic stiffness of the reinforcements respects
I m r
−1
( L - L ) : T
m
r I v
L = L +
(26)
m
1-
v