5.3 Prandtl Hardening Model
265
int A :=
{σ p , σ hk } | |σ p + σ hk | − [σ y − σ hi ] < 0
,
(5.182)
whereas the yield surface, which in the present one-dimensional case collapses to
the two planes σ p + σ hk = ±[σ y − σ hi ], is defined as the boundary of the admissible
domain, i.e.
∂ A :=
{σ p , σ hi } | |σ p + σ hk | − [σ y − σ hi ] = 0
,
(5.183)
Collectively, the admissible domain in the {σ p , σ hi , σ hk }-space is characterized by
the yield condition
|σ p + σ hk | − [σ y − σ hi ] ≤ 0.
(5.184)
States in the interior int A of the admissible domain with |σ p + σ hk | < σ y − σ hi are
elastic, whereas states on the boundary ∂ A of the admissible domain with |σ p +
σ hk | = σ y − σ hi are plastic.
The corresponding dual dissipation potential π
∗ , as determined from a Legendre
transformation
π
∗
(σ p , σ hi , σ hk ) = max
˙
p ,˙ hi ,˙ hk
(5.185)
{σ p ˙
p + σ hi ˙
hi + σ hk ˙
hk − [σ y − H hi ] |˙ p | + H hi ˙
hi − K hk [˙ p − ˙
hk ]}
then reads with the stationarity conditions σ hi = −H hi and σ hk = −K hk (note the
minus signs)
π
∗
(σ p , σ hi , σ hk ) = I A (σ p , σ hi , σ hk ) :=
(5.186)
⎧
⎨
⎩
0
|σ p + σ hk | ≤ σ y − σ hi
for
∞
|σ p + σ hk | > σ y − σ hi
⎫
⎬
⎭
,
where I A denotes the indicator function of the admissible domain A in the {σ p , σ hi ,
σ hk }-space. The evolution laws (the associated flow rules) for the plastic and the
isotropic- and kinematic-hardening strains then follow as some sub-derivatives of
the dual dissipation potential with respect to their conjugated variables
˙
p (σ p , σ hi , σ hk ) ∈ d σ p π
∗
(σ p , σ hi , σ hk ) = d σ p I A (σ p , σ hi , σ hk ),
˙
hi (σ p , σ hi , σ hk ) ∈ d σ hi π
∗
(σ p , σ hi , σ hk ) = d σ hi I A (σ p , σ hi , σ hk ),
˙
hk (σ p , σ hi , σ hk ) ∈ d σ hk π
∗
(σ p , σ hi , σ hk ) = d σ hk I A (σ p , σ hi , σ hk ),
(5.187)