Considering a more general stress space with normal stresses and couple stresses, the
yield surface can be considered as a hypersphere in stress space with normal b
N to the
yield surface F defined as
b
N ¼
∂F
∂Σ
! b n, b v
½ Š
ð5:164Þ
where b n ¼
∂F
∂σ
f ij
jξ ij j and b v ¼
∂F
∂l
À1 m
ℓ
À1 b C ij
jξ ij j .
And the flow rules read
_
ε
pl
ij ¼ γ
∂F
∂σ ij
γ
f ij
jξ ij j
γb n
ð5:165aÞ
ℓ _
χ
pl
ij ¼ γ
∂F
∂ℓ
À1 m ij
γℓ
À1
b
C ij
jξ ij j
γb ν
ð5:165bÞ
γ is the consistency parameter which is defined from the loading/unloading
conditions and is related to the evolution of the generalized equivalent plastic strain
defined by the hardening law as follows:
_
α ¼
ffiffi ffi
2
3
r
γ
ð5:166Þ
The constitutive model is completed with the evolution equations for the backstresses:
_
β ij ¼
2
3
H
0
γb n ij
ð5:167aÞ
ℓ _
η ij ¼
2
3
H
0
γb ν ij
ð5:167bÞ
where H
0 represents a kinematic hardening modulus which may be a linear or a
nonlinear function of the hardening parameter α. For instance, the assumption of a
constant kinematic hardening modulus leads to the so-called Prager-Ziegler rule,
(Fung and Tong 2001). Using Eqs. (5.165a) and (5.165b) into the generalized strain
norm for the plastic quantities yields
j _
Ε
pl j ¼ _
ε
pl
ij _
ε
pl
ij þ ℓ
2
_
χ
pl
ij _
χ
pl
ij
h
i 1=2
γ
jξ ij j
f ij f ij þ ℓ
À2 b
C ij b
C ij
h
i 1=2
ð5:168Þ
which implies j _
Ε
pl j ¼ γ.
Using this result in Eq. (5.166) and integrating yields
258
5 Unified Mechanics of Thermo-mechanical Analysis