17 Microplane Modeling for Inelastic Responses …
313
C
e(1)
=
1
E
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 −ν −ν
0
0
0
−ν 1 −ν
0
0
0
−ν −ν 1
0
0
0
0 0 0 2(1 + ν)
0
0
0 0 0
0
2(1 + ν)
0
0 0 0
0
0
2(1 + ν)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(17.38)
C
e(2)
i j
=
⎧
⎨
⎩
1
2 ¯
σ
β i
3Σ j − Σ m
1
E M
−
1
E A
∂ξ
∂ ¯
σ
j ≤ 3
3
¯
σ
β i Σ j
1
E M
−
1
E A
∂ξ
∂ ¯
σ
j > 3
(17.39)
C
tr(1)
i j
=
3
2π ¯
σ
ε
∗ ∂ξ s
∂ ¯
σ
⎧
⎪ ⎨
⎪ ⎩
Ω
Q i
2 A
3Σ j − Σ m
j ≤ 3
Ω
3Q i
A
Σ j
j < 3
(17.40)
C
tr(2)
i j
=
3
2π
ε
∗
ξ s
Ω
1
A
∂ Q i
∂Σ j
−
Q i
A
∂ A
∂Σ j
dΩ
(17.41)
(17.42)
∂ A
∂Σ
=
1
A
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
(Σ 1 N 1 + Σ 4 N 4 + Σ 5 N 5 − N 1 Σ N )
(Σ 2 N 2 + Σ 4 N 4 + Σ 6 N 6 − N 2 Σ N )
(Σ 3 N 3 + Σ 5 N 5 + Σ 6 N 6 − N 3 Σ N )
Σ 4 (N 1 + N 2 ) + N 4 (Σ 1 + Σ 2 ) + Σ 5 N 6 + Σ 6 N 5 − 2N 4 Σ N
Σ 5 (N 1 + N 3 ) + N 5 (Σ 1 + Σ 3 ) + Σ 4 N 6 + Σ 6 N 4 − 2N 5 Σ N
Σ 6 (N 2 + N 3 ) + N 6 (Σ 2 + Σ 3 ) + Σ 4 N 5 + Σ 5 N 4 − 2N 6 Σ N
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(17.43)
Using forward Euler implementation scheme, the stress vector at the end of an
increment can be obtained as:
(n+1)
=
(n)
+
(n)
=
(n)
+
C
(n)
−1
(n)
−
∂E
(n)
∂ T
(17.44)
As it can be seen, numerical integration over the surface of unit hemisphere is
needed for the implementation. Various techniques have been so far proposed for
numerical calculation of such integrals, and efficiency as well as error percentage of
various schemes have been evaluated in several works (Jalalpour et al. 2019; Huang
et al. 2017; Verron 2015; Badel and Leblond 2003; Nˇ emeˇ cek et al. 2002). Considering