312
6 Visco-Plasticity
Thereby the viscous and plastic contributions π v and π p to the (total) dissipation
potential π are chosen as
π v (˙ vp ) =
1
2
η |˙ vp |
2 and π p (˙ vp ) = σ y |˙ vp |.
(6.69)
Observe that (i) π = π v + π p does not depend on ˙
, thus the dissipative stress σ
=
σ − σ
≡ 0 vanishes identically, and that (ii) π v is positively homogenous of degree
two in ˙
vp and obviously smooth at the origin ˙
vp = 0, and that (iii) π p is positively
homogenous of degree one in ˙
vp and obviously non-smooth at the origin ˙
vp = 0.
As a consequence of the additive structure of the (total) dissipation potential the
dissipative visco-plastic stress σ
vp consists of a viscous contribution, the dissipative
viscous overstress σ
v , and a plastic contribution, the dissipative plastic stress σ
p , i.e.
σ
vp (˙ vp ) = σ
v (˙ vp ) + σ
p (˙ vp ).
(6.70)
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 ,
(6.71)
whereas the dissipative plastic stress σ
p computes as some sub-derivative of the
plastic dissipation potential with respect to its conjugated variable
σ
p (˙ vp ) ∈ d ˙
vp π p (˙ vp ) =
⎧
⎨
⎩
+σ y
˙
vp > 0
[−σ y , +σ y ] for ˙
vp = 0
−σ y
˙
vp < 0
⎫
⎬
⎭
,
(6.72)
whereby d ˙
vp π p denotes the set of sub-derivatives, i.e. the sub-differential of π p with
respect to ˙
vp .
Recall that the energetic and the dissipative visco-plastic stresses are constitutively
related by σ
vp + σ
vp = 0, thus the notion of visco-plastic stress together with the
notions of viscous overstress and plastic 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
(6.73)
will exclusively be used in the sequel for convenience of exposition.
Separate Viscous Overstress and Plastic Stress
The Perzyna 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.
The closed and convex admissible domain A = int A ∪ ∂ A in the space of the
(plastic) dissipative driving force, i.e. in the σ p -space, is next introduced as the
6 Visco-Plasticity
Thereby the viscous and plastic contributions π v and π p to the (total) dissipation
potential π are chosen as
π v (˙ vp ) =
1
2
η |˙ vp |
2 and π p (˙ vp ) = σ y |˙ vp |.
(6.69)
Observe that (i) π = π v + π p does not depend on ˙
, thus the dissipative stress σ
=
σ − σ
≡ 0 vanishes identically, and that (ii) π v is positively homogenous of degree
two in ˙
vp and obviously smooth at the origin ˙
vp = 0, and that (iii) π p is positively
homogenous of degree one in ˙
vp and obviously non-smooth at the origin ˙
vp = 0.
As a consequence of the additive structure of the (total) dissipation potential the
dissipative visco-plastic stress σ
vp consists of a viscous contribution, the dissipative
viscous overstress σ
v , and a plastic contribution, the dissipative plastic stress σ
p , i.e.
σ
vp (˙ vp ) = σ
v (˙ vp ) + σ
p (˙ vp ).
(6.70)
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 ,
(6.71)
whereas the dissipative plastic stress σ
p computes as some sub-derivative of the
plastic dissipation potential with respect to its conjugated variable
σ
p (˙ vp ) ∈ d ˙
vp π p (˙ vp ) =
⎧
⎨
⎩
+σ y
˙
vp > 0
[−σ y , +σ y ] for ˙
vp = 0
−σ y
˙
vp < 0
⎫
⎬
⎭
,
(6.72)
whereby d ˙
vp π p denotes the set of sub-derivatives, i.e. the sub-differential of π p with
respect to ˙
vp .
Recall that the energetic and the dissipative visco-plastic stresses are constitutively
related by σ
vp + σ
vp = 0, thus the notion of visco-plastic stress together with the
notions of viscous overstress and plastic 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
(6.73)
will exclusively be used in the sequel for convenience of exposition.
Separate Viscous Overstress and Plastic Stress
The Perzyna 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.
The closed and convex admissible domain A = int A ∪ ∂ A in the space of the
(plastic) dissipative driving force, i.e. in the σ p -space, is next introduced as the