62
4 Nonlinear Constitutive Relations
Fig. 4.3 Orientation of
reinforcement fibers
θ
Θ
1
˘
Θ
1
Θ
2
˘
Θ
2
˘ i
1
R e in fo r c e m e n t fi b r e s
i
1
˘ i
2 i
2
c = ˘
c
abcd ˘
i a ⊗ ˘
i b ⊗ ˘
i c ⊗ ˘
i d
= c
i jkl g i ⊗ g j ⊗ g k ⊗ g l ,
(4.22)
which leads to
c
i jkl
=
g
i
· ˘
i a
g
j
· ˘
i b
g
k
· ˘ i c
g
l
· ˘
i d
˘
c
abcd
.
(4.23)
Here, the indices, a, b, c and d, have the same function as i, j, k and l, but they
are used for the components in material coordinate system. Using the same rule one
obtains
˘
ε ab =
g
i
· ˘
i a
g
j
· ˘
i b
ε i j ,
(4.24)
σ
i j
=
g
i
· ˘
i a
g
j
· ˘
i b
˘
σ
ab
,
(4.25)
˘
E a =
g
i
· ˘
i a
E i ,
(4.26)
D
i
=
g
i
· ˘
i a
˘
D
a
,
(4.27)
which can be expressed in matrix form as
˘
ε = T ε , σ = T
T
˘
σ ,
(4.28)
˘
E = Q E , D = Q
T ˘
D .
(4.29)
Due to the neglect of the transverse normal strain ˘
ε 33 , the constitutive equations
given in (4.15) and (4.16) are simplified to contain only five components, which can
be expressed in matrix form as
˘
σ = ˘
c˘ ε − ˘
e
T ˘
E ,
(4.30)
˘
D = ˘
e ˘
ε + ˘
χ ˘
E ,
(4.31)