356
6 Visco-Plasticity
contribution, the dissipative viscous overstress σ
v , and a plastic contribution, the
dissipative plastic stress σ
p , i.e.
σ
vp (˙ vp , ˙
hk ) = σ
v (˙ vp ) + σ
p (˙ vp , ˙
hk ).
(6.169)
Thereby the dissipative viscous overstress σ
v computes as partial derivative of the
viscous dissipation potential with respect to its conjugated variable
σ
v (˙ vp ) = ∂ ˙
vp π v (˙ vp ) = η ˙
vp ,
(6.170)
whereas the dissipative plastic stress σ
p and the dissipative kinematic-hardening
stress σ
hk compute as some sub-derivatives of the plastic dissipation potential with
respect to their conjugated variables
σ
p (˙ vp , ˙
hk ) ∈ d ˙
vp π(˙ vp , ˙
hk ),
σ
hi (˙ vp , ˙
hk ) ∈ d ˙
hk π(˙ vp , ˙
hk ),
(6.171)
with
d ˙
vp π p (˙ vp , ˙
hk ) =
⎧
⎨
⎩
+[σ y + K hk ]
˙
vp > 0
−[σ y − K hk ], +[σ y + K hi ]
for ˙
vp = 0
−[σ y − K hi ]
˙
vp < 0
⎫
⎬
⎭
,
d ˙
hk π p (˙ vp , ˙
hk ) =
− K hk ,
(6.172)
whereby d ˙
vp π p and d ˙
hk π p denote the sets of sub-derivatives, i.e. the sub-differentials
of π p with respect to ˙
vp and ˙
hk , respectively.
Recall that the energetic and the dissipative visco-plastic as well a the kinematichardening stresses are constitutively related by σ
vp + σ
vp = 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 kinematic-hardening stress defined
as the values
σ vp = σ v + σ p := σ
vp = −σ
vp with σ v := σ
v and σ p := σ
p ,
(6.173a)
σ hk
:= σ
hk = −σ
hk ,
(6.173b)
will exclusively be used in the sequel for convenience of exposition.
Separate Viscous Overstress and Plastic Stress
The Perzyna kinematic hardening model may be formulated further by considering
the viscous overstress σ v and the plastic stress σ p separately. Thereby, due to the
non-smooth plastic dissipation potential, the plastic stress is constrained to reside in
an admissible domain.