Substitution of Eq. (5.19) into Eq. (5.18) yields for any value of the integration
parameter β between 0 and 1:
dX
Φ
nþ1 ¼ a nþ1 Δγ
S nþ1 À X
Φ
nþ1
S nþ1 À X
Φ
nþ1
À
c
0
2
c 1
X
Φ
n
!
ð5:20Þ
with a nþ1 ¼
c 1 1ÀΦ
ð
Þ
1þc 0
2
1ÀΦ
ð
Þ1Àβ ð
ÞΔγ .
Using the flow rule expressed in Eq. (5.3) allows us to express Eq. (5.17) as
S nþ1 ¼ S
tr
nþ1 À Δγ 1 À Φ
ð
Þ2μ
S nþ1 À X
Φ
nþ1
S nþ1 À X
Φ
nþ1
ð5:21Þ
And introducing the relative stress tensor in ξ
D
nþ1 ¼ S nþ1 À X
Φ
nþ1 yields
ξ
Φ
nþ1 ¼ S nþ1 À X
Φ
nþ1 S
tr
nþ1 À Δγ 1 À Φ
ð
Þ2μ
S nþ1 À X
Φ
nþ1
S nþ1 À X
Φ
nþ1
À X
Φ
n À dX
Φ
nþ1 ð5:22Þ
Using Eq. (5.20) into Eq. (5.22) gives
S nþ1 À X
Φ
nþ1 þ Δγ 1 À Φ
ð
Þ2μ þ a nþ1
½
Š
S nþ1 À X
Φ
nþ1
S nþ1 À X
Φ
nþ1
¼ B n
ð5:23Þ
which results after letting B n ¼ S
tr
nþ1 À X
Φ
n þ b nþ1 ΔγX n and b nþ1 ¼
c
0
2
c 1
a nþ1 .
The normal to the yield surface can be expressed in terms of the initial values of
the stress; the state variables and the strain increment at each step are as follows:
_
n nþ1
S nþ1 À X
Φ
nþ1
S nþ1 À X
Φ
nþ1
¼
B n
B n
k k
ð5:24Þ
Taking the trace product of Eq. (5.22) with itself yields
S nþ1 À X
Φ
nþ1
þ Δγ 1 À Φ
ð
Þ2μ þ a nþ1
½
Š
¼ S n À X
Φ
n
2 þ 1 À Φ
ð
Þ2μΔe nþ1 þ b nþ1 ΔγX
Φ
n
2
n
þ2 S n À X
Φ
n
À
Á : 1 À Φ
ð
Þ2μΔe nþ1 þ b nþ1 ΔγX
Φ
n
Â
Ã
'
:
1=2
ð5:25Þ
210
5 Unified Mechanics of Thermo-mechanical Analysis