ξ nþ1 ¼ Ξ Δγ
ð Þ
1
1 þ
2
3 H
0
Δγ 1 À Φ
ð
Þ
M
À1
ξ
tr
nþ1
ð5:228Þ
where
Ξ Δγ
ð Þ ¼ M
À1
þ
ΔγP
1 þ
2
3 H
0 1 À Φ
ð
ÞΔγ
"
# À1
ð5:229Þ
To complete the above algorithm, it is still necessary to compute the consistency
parameter Δγ, which can be obtained from the yield condition and the corresponding
constitutive model. Thus
F Δγ
ð Þ À Θ
ηΔγ
Δt
b f nþ1 À
ffiffi ffi
2
3
r
K α nþ1
ð
ÞÀΘ
ηΔγ
Δt
¼ 0
ð5:230Þ
where b f nþ1 = ξ
T
nþ1 Pξ nþ1
Â
à 1=2 and Θ
ηΔγ
Δt
À Á = φ
2 1 ηΔγ
Δt
À Á
as defined in the previous
section.
Equation (5.230) is a scalar nonlinear equation in the consistency parameter Δγ,
which can be solved by a local Newton iteration. In the Newton-Raphson iteration
scheme, the elastoplastic tangent modulus consistent with the integration scheme is
needed in order to preserve the convergence properties of the Newton algorithm.
Linearizing the above set of algorithmic equations yields
dΣ nþ1
dΕ nþ1
¼ 1 À Φ
ð
Þ Ξ Δγ
ð Þ À
1
1 þ e β
Ν Ν
0
@
1
A
ð5:231Þ
where
Ν ¼
Ξ Δγ
ð ÞPξ nþ1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ξ
T
nþ1 PΞ Δγ
ð ÞPξ nþ1
q
ð5:232Þ
e β ¼ 1 À Φ
ð
Þ
2θ 1
3θ 2
b f
2
nþ1 K
0
θ 1 þ H
0
θ 2
ð
Þþ
θ
2
1
θ 2
dΘ
dΔγ
b f nþ1
!
1
ξ
T
nþ1 PΞ Δγ
ð ÞPξ nþ1
ð5:233Þ
θ 1 ¼ 1 þ
2
3
H
0 1 À Φ
ð
ÞΔγ
θ 2 ¼ 1 À
2
3
K
0 1 À Φ
ð
ÞΔγ
270
5 Unified Mechanics of Thermo-mechanical Analysis
ð Þ
1
1 þ
2
3 H
0
Δγ 1 À Φ
ð
Þ
M
À1
ξ
tr
nþ1
ð5:228Þ
where
Ξ Δγ
ð Þ ¼ M
À1
þ
ΔγP
1 þ
2
3 H
0 1 À Φ
ð
ÞΔγ
"
# À1
ð5:229Þ
To complete the above algorithm, it is still necessary to compute the consistency
parameter Δγ, which can be obtained from the yield condition and the corresponding
constitutive model. Thus
F Δγ
ð Þ À Θ
ηΔγ
Δt
b f nþ1 À
ffiffi ffi
2
3
r
K α nþ1
ð
ÞÀΘ
ηΔγ
Δt
¼ 0
ð5:230Þ
where b f nþ1 = ξ
T
nþ1 Pξ nþ1
Â
à 1=2 and Θ
ηΔγ
Δt
À Á = φ
2 1 ηΔγ
Δt
À Á
as defined in the previous
section.
Equation (5.230) is a scalar nonlinear equation in the consistency parameter Δγ,
which can be solved by a local Newton iteration. In the Newton-Raphson iteration
scheme, the elastoplastic tangent modulus consistent with the integration scheme is
needed in order to preserve the convergence properties of the Newton algorithm.
Linearizing the above set of algorithmic equations yields
dΣ nþ1
dΕ nþ1
¼ 1 À Φ
ð
Þ Ξ Δγ
ð Þ À
1
1 þ e β
Ν Ν
0
@
1
A
ð5:231Þ
where
Ν ¼
Ξ Δγ
ð ÞPξ nþ1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ξ
T
nþ1 PΞ Δγ
ð ÞPξ nþ1
q
ð5:232Þ
e β ¼ 1 À Φ
ð
Þ
2θ 1
3θ 2
b f
2
nþ1 K
0
θ 1 þ H
0
θ 2
ð
Þþ
θ
2
1
θ 2
dΘ
dΔγ
b f nþ1
!
1
ξ
T
nþ1 PΞ Δγ
ð ÞPξ nþ1
ð5:233Þ
θ 1 ¼ 1 þ
2
3
H
0 1 À Φ
ð
ÞΔγ
θ 2 ¼ 1 À
2
3
K
0 1 À Φ
ð
ÞΔγ
270
5 Unified Mechanics of Thermo-mechanical Analysis