73
3
This is written in abbreviated form as
ε
σ
p
p q q
S
= ∑
(3.33)
σ
ε
p
p q q
C
= ∑
(3.34)
where ε p and σ p represent six independent components of strain
and stress, respectively. p and q run from 1 to 6. This can be shown
by converting ij to p and kl to q according to the following scheme.
Tensor notation
Matrix notation
11 22 33 23 32 13 31 12 21
1 2 3 4 5 6
Therefore, the first line of Eq. 3.28 in the contracted notation may be
written as
σ
ε
ε
ε
ε
ε
ε
xx
xx
yy
zz
XZ
YZ
XY
C
C
C
C
C
C
=
+
+
+
+
+
11
12
13
14
15
16
(3.35)
In line with Eq. 3.35, expressions for other stress components can be
derived at in terms of the stiffness matrix terms and strains. Thus,
σ
σ
σ
τ
τ
τ
1
2
3
1
2
3
11
12
13
14
15
16
12
22
=
C
C
C
C
C
C
C
C
C 2 23
24
25
26
13
14
15
16
23
24
25
26
33
34
35
36
34
44
45
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C 4 46
35
45
55
56
36
46
56
66
1
2
3
1
2
C
C
C
C
C
C
C
C
Y
Y
Y
ε
ε
ε
3 3
(3.36)
3.2.2 Orthotropy in Composites
A composite lamina is orthotropic in nature (i.e. it has symmetrical
structure in three principal planes). Its elastic properties are defined
by nine independent elastic constants. The thickness of lamina is
much smaller than other two dimensions (i.e. length and width).
For simplicity, the stresses normal to the plane of the lamina are
ignored. This is commonly referred to as the plane stress condition
(i.e. σ 3 = σ 13 = σ 23 = 0). However, strains are present when normal to
the lamina. The stress-strain relation can be written as
σ
σ
τ
ε
ε
τ
1
2
12
11
12
12
22
66
1
2
12
0
0
0
0
=
C
C
C
C
C
(3.37)
where four independent elastic constants (modulus and Poisson’s
ratio) of lamina are E 1 , E 2 , ν 12 , and ν 21 are given as
C
E
11
1
12 21
1
= −ν ν
(3.38)
3.2 · Macromechanics of Polymeric Composites
3
This is written in abbreviated form as
ε
σ
p
p q q
S
= ∑
(3.33)
σ
ε
p
p q q
C
= ∑
(3.34)
where ε p and σ p represent six independent components of strain
and stress, respectively. p and q run from 1 to 6. This can be shown
by converting ij to p and kl to q according to the following scheme.
Tensor notation
Matrix notation
11 22 33 23 32 13 31 12 21
1 2 3 4 5 6
Therefore, the first line of Eq. 3.28 in the contracted notation may be
written as
σ
ε
ε
ε
ε
ε
ε
xx
xx
yy
zz
XZ
YZ
XY
C
C
C
C
C
C
=
+
+
+
+
+
11
12
13
14
15
16
(3.35)
In line with Eq. 3.35, expressions for other stress components can be
derived at in terms of the stiffness matrix terms and strains. Thus,
σ
σ
σ
τ
τ
τ
1
2
3
1
2
3
11
12
13
14
15
16
12
22
=
C
C
C
C
C
C
C
C
C 2 23
24
25
26
13
14
15
16
23
24
25
26
33
34
35
36
34
44
45
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C 4 46
35
45
55
56
36
46
56
66
1
2
3
1
2
C
C
C
C
C
C
C
C
Y
Y
Y
ε
ε
ε
3 3
(3.36)
3.2.2 Orthotropy in Composites
A composite lamina is orthotropic in nature (i.e. it has symmetrical
structure in three principal planes). Its elastic properties are defined
by nine independent elastic constants. The thickness of lamina is
much smaller than other two dimensions (i.e. length and width).
For simplicity, the stresses normal to the plane of the lamina are
ignored. This is commonly referred to as the plane stress condition
(i.e. σ 3 = σ 13 = σ 23 = 0). However, strains are present when normal to
the lamina. The stress-strain relation can be written as
σ
σ
τ
ε
ε
τ
1
2
12
11
12
12
22
66
1
2
12
0
0
0
0
=
C
C
C
C
C
(3.37)
where four independent elastic constants (modulus and Poisson’s
ratio) of lamina are E 1 , E 2 , ν 12 , and ν 21 are given as
C
E
11
1
12 21
1
= −ν ν
(3.38)
3.2 · Macromechanics of Polymeric Composites