d b f nþ1
dΔγ
=
b g nþ1
b f nþ1
ð5:247Þ
where
b g nþ1 ¼ ξ
tr
T
nþ1 QG Δγ
ð ÞQ
T PQΓ Δγ
ð ÞQ
T
ξ
tr
nþ1
ð5:248Þ
G Δγ
ð Þ ¼
dΓ Δγ
ð Þ
dΔγ
¼ DIAG G 11 , G 22, G 33 , G 44 , G 55 , G 66
½
Š
with
G 11 ¼ G 22 ¼ G 44 ¼ G 55 ¼ G 66 ¼ À
2
3 1 À Φ
ð
ÞH
0
þ 2μ
Â
Ã
1 þ
2
3 1 À Φ
ð
ÞH
0
þ 2μ
Â
à Δγ
È
É 2
G 33 ¼ À
2
3 1 À Φ
ð
ÞH
0
1 þ
2
3 1 À Φ
ð
ÞH
0
Δγ
Â
à 2
Using the definition of K(α), Eq. (5.247), and θ 2 from Eq. (5.231) yields the
Jacobian needed for the local Newton iteration
Table 5.7 Return mapping integration algorithm-strain gradient model
Update strain
Ε n + 1 = Ε n + — Δu
Compute trial state
Σ
tr
nþ1 = 1 2 Φ
ð
ÞM Ε nþ1 2 Ε
vp
n 2 Ε
θ
nþ1
À
Á
ξ
tr
nþ1 = Σ
tr
nþ1 2 X n
Compute trial yield function
F
tr
nþ1 =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ξ
trT
nþ1 Pξ
tr
nþ1
q
2
ffiffi
2
3
q
K α n
ð Þ
IF F
tr
nþ1 > 0 THEN
Call Newton local and solve f(Δγ) = 0 for Δγ
Compute
Ξ Δγ
ð Þ= M
2 1 þ
ΔγP
1þ
2
3 H
0 1 2 Φ
ð
Þ Δγ
h
i 2 1
Update
ξ nþ1 = Ξ Δγ
ð Þ
1
1þ
2
3 H
0 1 2 Φ
ð
Þ Δγ
M
2 1 ξ
tr
nþ1
X nþ1 = X n þ Δγ
2
3 H
0 1 2 Φ
ð
Þξ nþ1
Σ n + 1 = ξ n + 1 + X n + 1
α nþ1 = α n þ Δγ 1 2 Φ
ð
Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2
3 ξ
T
nþ1 Pξ nþ1
q
Ε
vp
nþ1 = Ε
vp
n þ Δγ
Pξ nþ1
1 2 Φ
ð
Þ
Ε
e
nþ1 = Ε nþ1 2 Ε
vp
nþ1 2 Ε
θ
nþ1
Compute consistent Jacobian
dΣnþ1
dΕnþ1 = 1 2 Φ
ð
Þ Ξ Δγ
ð Þ2
1
1þ e β
À ÁΝ Ν
"
#
ELSE
Elastic step (EXIT)
END IF
EXIT
5.8 Cosserat Continuum Implementation in Unified Mechanics Theory
273