378
6 Visco-Plasticity
σ
p (˙ vp , ˙
hi , ˙
hk ) ∈ d ˙
vp π(˙ vp , ˙
hi , ˙
hk ),
σ
hi (˙ vp , ˙
hi , ˙
hk ) ∈ d ˙
hi π(˙ vp , ˙
hi , ˙
hk ),
σ
hi (˙ vp , ˙
hi , ˙
hk ) ∈ d ˙
hk π(˙ vp , ˙
hi , ˙
hk ),
(6.216)
with
d ˙
vp π p (˙ vp , ˙
hi , ˙
hk ) =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
+
[σ y + H hi ] + K hk
˙
vp > 0
−
[σ y + H hi ] − K hk
,
for ˙
vp = 0
+
[σ y + H hi ] + K hk
−
[σ y + H hi ] − K hk
˙
vp < 0
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
,
d ˙
hi π p (˙ vp , ˙
hi , ˙
hk ) =
− H hi ,
d ˙
hk π p (˙ vp , ˙
hi , ˙
hk ) =
− K hk .
(6.217)
whereby d ˙
vp π p , d ˙
hi π p and d ˙
hk π p denote the sets of sub-derivatives, i.e. the subdifferentials of π p with respect to ˙
vp , ˙
hi and ˙
hk , respectively.
Recall that the energetic and the dissipative visco-plastic as well a the isotropic- and kinematic-hardening stresses are constitutively related by σ
vp + σ
vp = 0,
σ
hi + σ
hi = 0 and σ
hk + σ
hk = 0, respectively, thus the notions of visco-plastic stress
(together with the notions of viscous overstress and plastic stress) as well as of
isotropic- and kinematic-hardening stresses defined as the values
σ vp = σ v + σ p := σ
vp = −σ
vp with σ v := σ
v and σ p := σ
p ,
(6.218a)
σ hi
:= σ
hi = −σ
hi ,
(6.218b)
σ hk
:= σ
hk = −σ
hk ,
(6.218c)
will exclusively be used in the sequel for convenience of exposition.
Separate Viscous Overstress and Plastic Stress
The Perzyna mixed hardening model may be formulated further by considering the
viscous overstress σ v and the plastic stress σ p separately. Thereby, due to the nonsmooth plastic dissipation potential, the plastic stress is constrained to reside in an
admissible domain.
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 hi , σ hk }-space, is next introduced as the
union of the elastic domain and the yield surface, compare the representation in Fig.
5.27. Thereby, the admissible domain may either be determined directly from the
expression of the sub-differential d ˙
vp π p in Eq. 6.217, or, alternatively, from evaluating
the formal definition of the sub-differential
6 Visco-Plasticity
σ
p (˙ vp , ˙
hi , ˙
hk ) ∈ d ˙
vp π(˙ vp , ˙
hi , ˙
hk ),
σ
hi (˙ vp , ˙
hi , ˙
hk ) ∈ d ˙
hi π(˙ vp , ˙
hi , ˙
hk ),
σ
hi (˙ vp , ˙
hi , ˙
hk ) ∈ d ˙
hk π(˙ vp , ˙
hi , ˙
hk ),
(6.216)
with
d ˙
vp π p (˙ vp , ˙
hi , ˙
hk ) =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
+
[σ y + H hi ] + K hk
˙
vp > 0
−
[σ y + H hi ] − K hk
,
for ˙
vp = 0
+
[σ y + H hi ] + K hk
−
[σ y + H hi ] − K hk
˙
vp < 0
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
,
d ˙
hi π p (˙ vp , ˙
hi , ˙
hk ) =
− H hi ,
d ˙
hk π p (˙ vp , ˙
hi , ˙
hk ) =
− K hk .
(6.217)
whereby d ˙
vp π p , d ˙
hi π p and d ˙
hk π p denote the sets of sub-derivatives, i.e. the subdifferentials of π p with respect to ˙
vp , ˙
hi and ˙
hk , respectively.
Recall that the energetic and the dissipative visco-plastic as well a the isotropic- and kinematic-hardening stresses are constitutively related by σ
vp + σ
vp = 0,
σ
hi + σ
hi = 0 and σ
hk + σ
hk = 0, respectively, thus the notions of visco-plastic stress
(together with the notions of viscous overstress and plastic stress) as well as of
isotropic- and kinematic-hardening stresses defined as the values
σ vp = σ v + σ p := σ
vp = −σ
vp with σ v := σ
v and σ p := σ
p ,
(6.218a)
σ hi
:= σ
hi = −σ
hi ,
(6.218b)
σ hk
:= σ
hk = −σ
hk ,
(6.218c)
will exclusively be used in the sequel for convenience of exposition.
Separate Viscous Overstress and Plastic Stress
The Perzyna mixed hardening model may be formulated further by considering the
viscous overstress σ v and the plastic stress σ p separately. Thereby, due to the nonsmooth plastic dissipation potential, the plastic stress is constrained to reside in an
admissible domain.
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 hi , σ hk }-space, is next introduced as the
union of the elastic domain and the yield surface, compare the representation in Fig.
5.27. Thereby, the admissible domain may either be determined directly from the
expression of the sub-differential d ˙
vp π p in Eq. 6.217, or, alternatively, from evaluating
the formal definition of the sub-differential