C ¼ κI I þ 2μ
c
a À
1
3
I I
and D = 2μ
c
a
ð5:181Þ
and after simplifying yields
M
ep
¼
κI I þ 2μ b
‘ À
1
3
I I À
b n b n
b
K
À2μ
b n b v
b
K
À2μ
b n b v
b
K
2μ b
‘ À
b v v
b
K
2
6
6
6
4
3
7
7
7
5
ð5:182Þ
Alternatively it can be written as
M
ep
¼
C À C :
b n C : b n
b
K
ÀC :
b n D : b v
b
K
ÀD :
b v C : b n
b
K
D À D :
b v D : b v
b
K
2
6
6
4
3
7
7
5
ð5:183Þ
The flow theory just described can be written in the following alternative form,
which is useful in the numerical treatment of the problem. Using the generalized
vector and matrix notation, we define for the simple case of isotropic hardening:
Σ ¼
σ
ℓ
À1 m
"
#
, Ε ¼
ε
ℓχ
" #
, M ¼
C 0
0 D
!
ð5:184Þ
Then the constitutive model equations can be written as Σ ¼ M : Ε
el
. The yield
surface and flow rule follow
F σ, ℓ
À1 m, α
À
Á ¼ jΣ
0
j À
ffiffi ffi
2
3
r
K α
ð Þ
ð5:185Þ
_
E
pl ¼ γ b
N
ð5:186Þ
then
_
F ¼
∂F
∂Σ
Á _
Σ þ
∂F
∂Ε
pl
Á _
Ε
pl
ð5:187Þ
Using Hooke’s law and the flow rule results in
_
F ¼ b
N : M : _
Ε À γN : M : b
N þ
∂F
∂Ε
pl
Á γ b
N
ð5:188Þ
Utilizing the consistency conditions γ _
F ¼ 0 gives
5.8 Cosserat Continuum Implementation in Unified Mechanics Theory
261