dσ nþ1 ¼ 1 À Φ
ð
Þ C À 2μb n nþ1
∂Δγ
∂ε nþ1
À 2μΔγ
∂b n nþ1
∂ε nþ1
!
: dε nþ1
ð5:33Þ
Ignoring differentiation with respect to entropy generation term in the thermodynamic state index rate is not correct. If the strain increments are small, then the
ignored part of the differentiation is smaller by an order of magnitude. However, our
objective here is to present an earlier and simpler version of the UMT formulation.
The operator represents the ordered (dyadic) outer product and b n nþ1 is the
normal vector to the yield surface as defined in Eq. (5.3) evaluated at the end of the
increment, and
∂Δγ
∂ε nþ1
can be calculated from Eq. (5.25):
∂Δγ
∂ε nþ1
¼
b n nþ1
K 3
ð5:34Þ
Using
K 3 ¼ K 1 þ K 2
K 1 ¼ 1 þ
K
0
3μ
þ
a nþ1
2μ 1 À Φ
ð
Þ
K 2 ¼
a
0
nþ1 Δγ
2μ 1 À Φ
ð
Þ
þ
b n nþ1 b nþ1
2μ 1 À Φ
ð
Þ
b nþ1 1 À β
ð
ÞΔγ À 1
½
Š : X
Φ
n þ
1
2μ 1 À Φ
ð
Þ
∂Θ
∂Δγ
In addition,
∂b n nþ1
∂ε nþ1
can be obtained from Eq. (5.24) as
∂b n nþ1
∂ε nþ1
¼
∂b n nþ1
∂B
!
n
∂B n
∂ε nþ1
1
B nþ1
k
k
b I À b n nþ1 b n nþ1
:
∂B n
∂ε nþ1
ð5:35aÞ
and
∂B n
∂ε nþ1
¼ 2μ 1 À Φ
ð
Þ
c
Y À
1
3
I I
þ b
0
nþ1 Δγ þ b nþ1
À
Á
X n
∂Δγ
∂ε nþ1
where b
Q
is a fourth-order unit tensor mapped into the unit matrix and I a secondorder identity tensor mapped into unit column vector and K
0 denotes the derivative
with respect to the argument in K(α).
Letting
K 4 ¼ b
0
nþ1 Δγ þ b nþ1 and substituting
∂B n
∂ε nþ1
in Eq. (5.35a) yields
212
5 Unified Mechanics of Thermo-mechanical Analysis