312
M. R. Karamooz-Ravari et al.
Using some algebraic calculations and simplifications, the utilized vectors P, ˆ
N,
and R are obtained as:
P =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
Σ
2
1 + Σ
2
4 + Σ
2
5
Σ
2
2 + Σ
2
4 + Σ
2
6
Σ
2
3 + Σ
2
5 + Σ
2
6
Σ 4 (Σ 1 + Σ 2 ) + Σ 5 Σ 6
Σ 5 (Σ 1 + Σ 3 ) + Σ 4 Σ 6
Σ 6 (Σ 2 + Σ 3 ) + Σ 4 Σ 5
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(17.32)
ˆ
N =
N 1 N 2 N 3 2N 4 2N 5 2N 6
T
(17.33)
R =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
(Σ 1 N 1 + Σ 4 N 4 + Σ 5 N 5 )
(Σ 2 N 2 + Σ 4 N 4 + Σ 6 N 6 )
(Σ 3 N 3 + Σ 5 N 5 + Σ 6 N 6 )
Σ 4 (N 1 + N 2 ) + N 4 (Σ 1 + Σ 2 ) + Σ 5 N 6 + Σ 6 N 5
Σ 5 (N 1 + N 3 ) + N 5 (Σ 1 + Σ 3 ) + Σ 4 N 6 + Σ 6 N 4
Σ 6 (N 2 + N 3 ) + N 6 (Σ 2 + Σ 3 ) + Σ 4 N 5 + Σ 5 N 4
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(17.34)
In the relations above, (·) denotes the dot product operator. Since E i = E i (, T ),
the incremental form of the constitutive equations can be expressed as:
E i =
6
j=1
∂ E i
∂Σ j
Σ j +
∂ E i
∂ T
T
(17.35)
where (∂) denote the partial differentiation,
∂ E i
∂Σ j
−1
the continuum tangent stiffness matrix and
∂E i
∂Σ j
−1 ∂E i
∂ T
the tangent thermal moduli vector. Differentiating
Eq. (17.25) yields:
∂E i
∂Σ j
= C i j = C
e(1)
i j + C
e(2)
i j + C
tr(1)
i j
+ C
tr(2)
i j
(17.36)
∂E i
∂ T
=
⎧
⎪ ⎨
⎪ ⎩
1
E M
−
1
E A
∂ξ
∂ T
[(1 + ν)Σ i − νΣ m ] +
3
4π
ε
∗ ∂ξ s
∂ T
Ω
Q i
A
dΩ i ≤ 3
2
1
E M
−
1
E A
∂ξ
∂ T (1 + ν)Σ i +
3
2π
ε
∗ ∂ξ s
∂ T
Ω
Q i
A
dΩ
i > 3
(17.37)
in which