Composite Materials Research Progress

26
Jacquemin Frédéric and Fréour Sylvain
plane tensors of hygroscopic expansion coefficients and moduli. Those tensors are assumed to
be material constants.
with,
I
⎧ I
σ ⎫ ⎡ I
L
11
⎪ ⎪ ⎢
11
I
⎪σ
⎪ ⎢ I
22 L12
⎨ ⎬ = ⎢
I
⎪σ33
⎪ ⎢ I
L12
⎪ I ⎪ ⎢
⎩τ12
⎭ ⎢
⎣0
⎪⎧
τ
⎨
⎪⎩ τ
I
32
I
13
I
L12
I
L22
I
L23
0
⎪⎫
⎡L
⎬ = ⎢
⎪⎭ ⎢⎣
0
I
L ⎤
12 0
⎥
I
L ⎥
23 0
⎥
I
L ⎥
22 0
⎥
I
0 L ⎥
55 ⎦
I
44
0
L
I
55
⎤
⎥
⎥⎦
⎧ε
⎪
⎪ε
⎨
⎪ε
⎪
⎩γ
⎪⎧
γ
⎨
⎪⎩ γ
I
11
I
13
I
22
I
33
I
12
I
32
⎪⎫
⎬
⎪⎭
− β
− β
− β
− β
I
11
I
22
I
12
I
ΔC
⎫
⎪
I
ΔC
⎪
⎬ I
ΔC
⎪
I ⎪
ΔC
⎭
I c
Δ C = . c
I
ρ
I and ρ I are respectively the macroscopic moisture concentration and the
mass density of the dry material.
To solve the hygromechanical problem, it is necessary to express the strains versus the
displacements along with the compatibility and equilibrium equations.
Introducing a characteristic modulus L 0 , we introduce the following dimensionless
variables:
σ
Ι
= σ
Ι I I I I I I I I
/ L0
, L = L /L0
, ( w , u , v ) = ( w , u , v ) / b.
Displacements with respect to longitudinal and circumferencial directions, respectively
u ( x,
r)
I
and v ( x,
r)
I
are then deduced:
⎧ I
u ( x,
r)
= R1x
⎪
I
⎨v
( x,
r)
= R 2xr
⎪
R1,
R 2 are constants.
⎩
It is worth noticing that the displacements ( x,
r)
and ( x,
r)
do not depend on the
moisture concentration field. Finally, to obtain the through-thickness or radial component of
the displacement
concentration (47).
I
w , we shall consider in the following the analytical transient
I
The radial component of the displacement field w satisfies the following equation:
u I
v I
I
22
(48)
(49)
(50)