Composite Materials Research Progress

24
Jacquemin Frédéric and Fréour Sylvain
m m m m m
( L11
+ 2L12
)( β11
ΔC + M11
ΔT)
I I I I I I I I I
−L12
( β11ΔC
+ M11ΔT)
− ( L22
+ L23
)( β22ΔC
+ M33ΔT)
I m I
( L12
− L12
) ε11
I m m I I m m I
I22 L22
( 5 L11
− L12
+ 3 L22
) − L23(
3 L11
+ L12
+ 4 L22
)
L
2 2
I I m m I I ( 3 L22
− L23
)( L11
− L12
) + L22
− L23
I m m I I m m I
I22 L22
( L11
− 5 L12
− L22
) + L23(
L11
+ 3 L12
+ 4 L22
) −
L
2 2
I I m m I I ( 3 L22
− L23
)( L11
− L12
) + L22
− L23
⎧ m
N
⎪
1 =
⎪ m
N2
=
⎪
⎪ m
N3
=
⎪
⎪
⎪ m
N =
⎨ 4
⎪
⎪
⎪
⎪ m
N5
=
⎪
⎪
⎪ m m m I I
⎩D1
= L11
+ L12
+ L22
− L23
I
2
+ L23
I
ε22
2
I
3 L23
I
ε33
The pseudo-macroscopic stress tensors are deduced from the strains using (32). Thus, in
the matrix, one will have:
with
⎧ m m
σ
⎪
11 = L11
m m
⎨σ22
= L11
⎪ m m
⎪σ
=
⎩ 33 L11
(40)
⎡ m m m m m
σ
⎤
⎢
11 2 L44ε12
2 L44ε13
m m m m m m ⎥
σ = ⎢2
L44ε12
σ22
2 L44ε
23 ⎥
(41)
⎢ m m m m m ⎥
⎢
2 L
⎣ 44ε13
2 L44ε
23 σ33
⎥⎦
m m m m m m m m m ( ε11
− M11
ΔT)
+ L12
( ε22
+ ε33
− 2 M11
ΔT)
− β11(
L11
+ 2 L12
)
m m m m m m m m m ( ε22
− M11
ΔT)
+ L12
( ε11
+ ε33
− 2 M11
ΔT)
− β11(
L11
+ 2 L12
)
m m m m m m m m m ( ε33
− M11
ΔT)
+ L12
( ε11
+ ε22
− 2 M11
ΔT)
− β11(
L11
+ 2 L12
)
m
ΔC
m
ΔC
m
ΔC
The local mechanical states in the fiber are provided by Hill’s strains and stresses average
laws (36):
ε
σ
(42)
m
r 1 I v m
= ε − ε
(43)
r r
v
v
m
r 1 I v m
= σ − σ
(44)
r r
v
v