Q 1 ¼
0
2=
ffiffi ffi
6
p
1=
ffiffi ffi
3
p
À
ffiffi ffi
2
p =2 À1=
ffiffi ffi
6
p
1=
ffiffi ffi
3
p
ffiffi ffi
2
p =2 À1=
ffiffi ffi
6
p
1=
ffiffi ffi
3
p
2
6
4
3
7
5
ð5:240bÞ
P ¼
P 1 0
0 P 2
!
ð5:241aÞ
with
P 1 ¼
2=3 À1=3 À1=3
À1=3 2=3 À1=3
À1=3 À1=3 2=3
2
6
4
3
7
5 and P 2 ¼ DIAG 2, 1, 1
½
Š
ð5:241bÞ
Λ P = DIAG 1, 1, 0, 2, 1, 1
½
Š and
Λ M = DIAG 2μ, 2μ, 3λ þ 2μ, μ, 2μ, 2μ
½
Š
Now the update formula (5.237) can be written as
ξ nþ1 ¼ 1 þ
2
3
ΔγH
0
I þ ΔγQΛ P Λ M Q
T
h
i À1
ξ
tr
nþ1
ð5:242Þ
or after letting
Γ Δγ
ð Þ ¼
1 þ
2
3
ΔγH
0 1 À Φ
ð
Þ
I þ ΔγΛ P Λ M
! À1
ð5:243Þ
this expression becomes
ξ nþ1 ¼ QΓ Δγ
ð ÞQ
T
ξ
tr
nþ1
ð5:244Þ
where explicitly
Γ Δγ
ð Þ=DIAG
1
1 þ
2
3
1 À Φ
ð
ÞH
0 þ 2μ
Δγ
,
1
1 þ
2
3
1 À Φ
ð
ÞH
0 þ 2μ
Δγ
,
1
1 þ
2
3
1 À Φ
ð
ÞH
0 Δγ
,
1
1 þ
2
3
1 À Φ
ð
ÞH
0 þ 2μ
Δγ
,
1
1 þ
2
3
1 À Φ
ð
ÞH
0 þ 2μ
Δγ
,
1
1 þ
2
3
1 À Φ
ð
ÞH
0 þ 2μ
Δγ
2
6
6
6
6
6
4
3
7
7
7
7
7
5
ð5:245Þ
Making use of Eq. (5.214), b f nþ1 becomes
b f nþ1 ¼ ξ
tr
T
nþ1 Q
T
Γ Δγ
ð ÞQPQ
T
Γ Δγ
ð Þξ
tr
nþ1
h
i 1=2
ð5:246Þ
Using Eqs. (5.244)–(5.246),
d b f nþ1
dΔγ can be defined as
272
5 Unified Mechanics of Thermo-mechanical Analysis