5.3 Prandtl Hardening Model
247
∂ A :=
{σ p , σ hk } | |σ p + σ hk | − σ y = 0
,
(5.148)
Collectively, the admissible domain in the {σ p , σ hk }-space is characterized by the
yield condition
|σ p + σ hk | − σ y ≤ 0.
(5.149)
States in the interior int A of the admissible domain with |σ p + σ hk | < σ y are elastic,
whereas states on the boundary ∂ A of the admissible domain with |σ p + σ hk | = σ y
are plastic.
The corresponding dual dissipation potential π
∗ , as determined from the Legendre
transformation
π
∗
(σ p , σ hk ) = max
˙
p ,˙ hk
{σ p ˙
p + σ hk ˙
hk − σ y |˙ p | − K hk [˙ p − ˙
hk ]}
(5.150)
then reads with the stationarity condition σ hk = −K hk (note the minus sign)
π
∗
(σ p , σ hk ) = I A (σ p , σ hk ) :=
⎧
⎨
⎩
0
|σ p + σ hk | ≤ σ y
for
∞
|σ p + σ hk | > σ y
⎫
⎬
⎭
,
(5.151)
where I A denotes the indicator function of the admissible domain A in the {σ p , σ hk }space. The evolution laws (the associated flow rules) for the plastic and the kinematichardening strains then follow as some sub-derivatives of the dual dissipation potential
with respect to their conjugated variables
˙
p (σ p , σ hk ) ∈ d σ p π
∗
(σ p , σ hk ) = d σ p I A (σ p , σ hk ),
˙
hk (σ p , σ hk ) ∈ d σ hk π
∗
(σ p , σ hk ) = d σ hk I A (σ p , σ hi ),
(5.152)
with
d σ p π
∗
(σ p , σ hk )
= d σ p I A (σ p , σ hk ) =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0
|σ p + σ hk | < σ y
for
λ
σ p + σ hk
|σ p + σ hk |
|σ p + σ hk | = σ y
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
(5.153a)
and
d σ hk π
∗
(σ p , σ hk )
= d σ hk I A (σ p , σ hh ) =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0
|σ p + σ hk | < σ y
for
λ
σ p + σ hk
|σ p + σ hk |
|σ p + σ hk | = σ y
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
,
(5.153b)