6.3 Perzyna Hardening Model
357
The closed and convex admissible domain A = int A ∪ ∂ A in the space of the
dissipative driving forces, i.e. in the {σ vp , σ hk }-space, is next introduced as the union
of the elastic domain and the yield surface, compare the representation in Fig. 5.20.
Thereby, the admissible domain may either be determined directly from the expression of the sub-differential d ˙
vp π p in Eq. 6.172, or, alternatively, from evaluating the
formal definition of the sub-differential
d ˙
vp π p (˙ vp , ˙
hk ) =
(6.174)
{σ p | σ p [˙
vp − ˙
vp ] ≤ σ y
|˙
vp | − |˙ vp |
+ K hk [˙
vp − ˙
vp ] ∀˙
vp },
whereby ˙
vp denotes any admissible visco-plastic strain rate. Then at ˙
vp = 0 it
holds for any admissible ˙
vp that σ p ˙
vp ≤ σ y |˙
vp | + K hk ˙
vp and, with max ˙
vp
{[σ p −
K hk ] ˙
vp /|˙
vp |} = |σ p − K hk |, the admissible domain follows as |σ p − K hk | ≤
σ y . Moreover, the sub-differential d ˙
hk π p reduces to the partial derivative ∂ ˙
hk π p
and renders σ hk = −K hk . Thus the admissible domain is eventually expressed as
|σ p + σ hk | ≤ σ y .
The elastic domain is defined as the interior of the admissible domain, i.e.
int A :=
{σ p , σ hk } | |σ p + σ hk | − σ y < 0
,
(6.175)
whereas the yield surface, which in the present one-dimensional case collapses to
the two lines σ p + σ hk = ±σ y , is defined as the boundary of the admissible domain,
i.e.
∂ A :=
{σ p , σ hk } | |σ p + σ hk | − σ y = 0
.
(6.176)
Collectively, the admissible domain in the {σ p , σ hk }-space is characterized by the
yield condition
|σ p + σ hk | − σ y ≤ 0.
(6.177)
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 visco-plastic.
The corresponding dual viscous and plastic dissipation potentials π
∗
v and π
∗
p , as
determined from the Legendre transformations
π
∗
v (σ v
) = max
˙
vp
σ v ˙
vp −
1
2
η |˙ vp |
2
(6.178a)
π
∗
p (σ p , σ hk ) = max
˙
vp ,˙ hk
{σ p ˙
vp + σ hk ˙
hk − σ y |˙ vp | − K hk [˙ vp − ˙
hk ]}
(6.178b)
then read with the stationarity condition σ hk = −K hk (note the minus sign)
357
The closed and convex admissible domain A = int A ∪ ∂ A in the space of the
dissipative driving forces, i.e. in the {σ vp , σ hk }-space, is next introduced as the union
of the elastic domain and the yield surface, compare the representation in Fig. 5.20.
Thereby, the admissible domain may either be determined directly from the expression of the sub-differential d ˙
vp π p in Eq. 6.172, or, alternatively, from evaluating the
formal definition of the sub-differential
d ˙
vp π p (˙ vp , ˙
hk ) =
(6.174)
{σ p | σ p [˙
vp − ˙
vp ] ≤ σ y
|˙
vp | − |˙ vp |
+ K hk [˙
vp − ˙
vp ] ∀˙
vp },
whereby ˙
vp denotes any admissible visco-plastic strain rate. Then at ˙
vp = 0 it
holds for any admissible ˙
vp that σ p ˙
vp ≤ σ y |˙
vp | + K hk ˙
vp and, with max ˙
vp
{[σ p −
K hk ] ˙
vp /|˙
vp |} = |σ p − K hk |, the admissible domain follows as |σ p − K hk | ≤
σ y . Moreover, the sub-differential d ˙
hk π p reduces to the partial derivative ∂ ˙
hk π p
and renders σ hk = −K hk . Thus the admissible domain is eventually expressed as
|σ p + σ hk | ≤ σ y .
The elastic domain is defined as the interior of the admissible domain, i.e.
int A :=
{σ p , σ hk } | |σ p + σ hk | − σ y < 0
,
(6.175)
whereas the yield surface, which in the present one-dimensional case collapses to
the two lines σ p + σ hk = ±σ y , is defined as the boundary of the admissible domain,
i.e.
∂ A :=
{σ p , σ hk } | |σ p + σ hk | − σ y = 0
.
(6.176)
Collectively, the admissible domain in the {σ p , σ hk }-space is characterized by the
yield condition
|σ p + σ hk | − σ y ≤ 0.
(6.177)
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 visco-plastic.
The corresponding dual viscous and plastic dissipation potentials π
∗
v and π
∗
p , as
determined from the Legendre transformations
π
∗
v (σ v
) = max
˙
vp
σ v ˙
vp −
1
2
η |˙ vp |
2
(6.178a)
π
∗
p (σ p , σ hk ) = max
˙
vp ,˙ hk
{σ p ˙
vp + σ hk ˙
hk − σ y |˙ vp | − K hk [˙ vp − ˙
hk ]}
(6.178b)
then read with the stationarity condition σ hk = −K hk (note the minus sign)