5.3 Prandtl Hardening Model
229
∂ A :=
{σ p , σ hi } | |σ p | − [σ y − σ hi ] = 0
.
(5.112)
Collectively, the admissible domain in the {σ p , σ hi }-space is characterized by the
yield condition
|σ p | − [σ y − σ hi ] ≤ 0.
(5.113)
States in the interior int A of the admissible domain with |σ p | < σ y − σ hi are elastic,
whereas states on the boundary ∂ A of the admissible domain with |σ p | = σ y − σ hi
are plastic.
The corresponding dual dissipation potential π
∗ , as determined from the Legendre
transformation
π
∗
(σ p , σ hi ) = max
˙
p ,˙ hi
{σ p ˙
p + σ hi ˙
hi − [σ y + H hi ] |˙ p | + H hi ˙
hi }
(5.114)
then reads with the stationarity condition σ hi = −H hi (note the minus sign)
π
∗
(σ p , σ hi ) = I A (σ p , σ hi ) :=
⎧
⎨
⎩
0
|σ p | ≤ σ y − σ hi
for
∞
|σ p | > σ y − σ hi
⎫
⎬
⎭
,
(5.115)
where I A denotes the indicator function of the admissible domain A in the {σ p , σ hi }space. The evolution laws (the associated flow rules) for the plastic and the isotropichardening strains then follow as some sub-derivatives of the dual dissipation potential
with respect to their conjugated variables
˙
p (σ p , σ hi ) ∈ d σ p π
∗
(σ p , σ hi ) = d σ p I A (σ p , σ hi ),
˙
hi (σ p , σ hi ) ∈ d σ hi π
∗
(σ p , σ hi ) = d σ hi I A (σ p , σ hi ),
(5.116)
with
d σ p π
∗
(σ p , σ hi )
= d σ p I A (σ p , σ hi ) =
⎧
⎪ ⎨
⎪ ⎩
0
|σ p | < σ y − σ hi
for
λ
σ p
|σ p |
|σ p | = σ y − σ hi
⎫
⎪ ⎬
⎪ ⎭
(5.117a)
and
d σ hi π
∗
(σ p , σ hi )
= d σ hi I A (σ p , σ hi ) =
⎧
⎨
⎩
0
|σ p | < σ y − σ hi
for
λ
|σ p | = σ y − σ hi
⎫
⎬
⎭
,
(5.117b)
whereby d σ p π
∗ and d σ hi π
∗ denote the sets of sub-derivatives, i.e. the sub-differentials
of π
∗ with respect to σ p and σ hi , respectively, and λ is a positive Lagrange (or rather
plastic) multiplier.