composites mechincal analyis of lamina hygrothermal stresses. composites mechincal analyis of lamina hygrothermal stresses. composites mechincal analyis of lamina hygrothermal stresses.composites mechincal analyis of lamina hygrothermal stresses.composites mechincal analyis of lamina hygrothermal stresses

1. Macromechanical Analysis of a Lamina
Hygrothermal Stresses and Strains

2.  For a unidirectional lamina
 Thermally induced strains:
 Moisture induced strains:




























































0
+
0
+
S
0
0
0
S
S
0
S
S
= C
2
C
1
T
2
T
1
12
2
1
66
22
12
12
11
12
2
1



































0
T
=
0
2
1
T
2
T
1





























0
C
=
0
2
1
C
2
C
1




(2.174)
(2.175)
(2.176)

3.  For a unidirectional lamina




























































0
0
0
0
0
0
2
1
2
1
12
2
1
66
22
12
12
11
12
2
1
ε
ε
+
ε
ε
+
τ
σ
σ
S
S
S
S
S
=
γ
ε
ε
C
C
T
T




































γ
ε
-
ε
-
ε
ε
-
ε
-
ε
Q
Q
Q
Q
Q
=
τ
σ
σ
C
T
C
T
12
2
2
2
1
1
1
66
22
12
12
11
12
2
1
0
0
0
0
(2.174)
(2.177)

4.  For an angular lamina
 Thermally induced strains:
 Moisture induced strains:
































































γ
ε
ε
+
γ
ε
ε
+
τ
σ
σ
S
S
S
S
S
S
S
S
S
=
γ
ε
ε
C
xy
C
y
C
x
T
xy
T
y
T
x
xy
y
x
xy
y
x
66
26
16
26
22
12
16
12
11

































xy
y
x
T
xy
T
y
T
x
T
=

































xy
y
x
C
xy
C
y
C
x
C
=
(2.178)
(2.179)
(2.180)

5.  For an angular lamina
























0
2
2
1
1
α
α
]
= [T
/
α
α
α
-
xy
y
x
















s
c
sc
sc
sc
c
s
sc
s
c
=
T
2
2
2
2
2
2
1
2
2
]
[












s
-
c
sc
sc
-
2sc
-
c
s
2sc
s
c
=
[T]
2
2
2
2
2
2
)
(
=
c 
Cos
)
(
=
s 
Sin
(2.181)
(2.95) (2.96)
(2.97a,b)

6.  For an angular lamina
























0
]
[T
=
/2
2
1
1
-
xy
y
x





















s
c
sc
sc
2sc
c
s
2sc
s
c
=
]
[T
2
2
2
2
2
2
1












s
-
c
sc
sc
-
2sc
-
c
s
2sc
s
c
=
[T]
2
2
2
2
2
2
)
(
=
c 
Cos
)
(
=
s 
Sin
(2.95) (2.96)
(2.97a,b)
(2.182)

7. Find the following for a 600
angle lamina of Glass/Epoxy
a) coefficients of thermal expansion,
b) coefficients of moisture expansion,
c) strains under a temperature change of -1000
C and a moisture absorption of 0.02 kg/kg.
Use properties of unidirectional Glass/Epoxy lamina from Table 2.1.

8. a) From Table 2.1,
C,
/
m/m
10
8.6
= 0
-6
1 

C.
/
m/m
10
22.1
= 0
-6
2 

Using Equation (2.181), gives
,
0
10
22.1
10
8.6
0.5000
-
0.4330
-
0.4330
0.8660
0.2500
0.7500
0.8660
-
0.7500
0.2500
=
/2
6
-
-6
xy
y
x







































C.
/
m/m
10
11.69
-
10
11.98
10
18.73
= 0
6
-
6
-
-6
xy
y
x






























.
0
]
[T
=
/2
2
1
1
-
xy
y
x





























(2.181)

9. b) From Table 2.1
m/m/kg/kg,
0
=
1

m/m/kg/kg.
0.6
=
2

Using Equation (2.182) gives
































0
0.6
0.0
0.5000
-
0.4330
-
0.4330
0.8660
0.2500
0.7500
0.8660
-
0.7500
0.2500
=
/2
xy
y
x



m/m/kg/kg
0.5196
-
0.1500
0.4500
=
xy
y
x

















































0
]
[T
=
/2
2
1
1
-
xy
y
x





(2.182)

10. c) Now using Equations (2.179) and (2.180) to calculate the strains as.
(0.02)
0.5196
-
0.1500
0.4500
+
(-100)
10
11.69
-
10
11.98
10
18.73
=
6
-
6
-
-6
xy
y
x








































m/m
10
0.9223
-
10
0.1802
10
0.7127
=
2
2
2



























































































C
xy
C
y
C
x
T
xy
T
y
T
x
xy
y
x
66
26
16
26
22
12
16
12
11
xy
y
x
+
+
S
S
S
S
S
S
S
S
S
= (2.178)