288
6 Visco-Plasticity
Recall that the total stress applied to the rheological model (that enters the equilibrium condition) and the energetic and the dissipative stresses are constitutively
related by σ = σ
+ σ
, thus (since here σ
≡ 0)
σ ≡ σ
,
(6.9)
moreover the notions of viscous overstress and plastic stress defined as the values
σ := σ v + σ p with σ v := σ
v and σ p := σ
p
(6.10)
will exclusively be used in the sequel for convenience of exposition.
Separate Viscous Overstress and Plastic Stress
The Bingham 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 union
of the rigid domain and the yield surface, compare the representation in Fig. 5.2.
Thereby, the admissible domain may either be determined directly from the expression of the sub-differential d ˙
π p in Eq. 6.8, or, alternatively, from evaluating the
formal definition of the sub-differential
d ˙
π p (˙ ) = {σ p | σ p [˙
− ˙
] ≤ σ y
|˙
| − |˙ |
∀˙
},
(6.11)
whereby ˙
denotes any admissible strain rate. Then at ˙
= 0 it holds for any admissible ˙
that σ p ˙
≤ σ y |˙
| and, with max ˙
{σ p ˙
/|˙
|} = |σ p |, the admissible domain
follows as |σ p | ≤ σ y .
The rigid domain is defined as the interior of the admissible domain, i.e.
int A := {σ p | |σ p | − σ y < 0},
(6.12)
whereas the yield surface, which in the present one-dimensional case collapses to
the two end points σ p = ±σ y , is defined as the boundary of the admissible domain,
i.e.
∂ A := {σ p | |σ p | − σ y = 0}.
(6.13)
Collectively, the admissible domain in the σ p -space is characterized by the yield
condition
|σ p | − σ y ≤ 0.
(6.14)
6 Visco-Plasticity
Recall that the total stress applied to the rheological model (that enters the equilibrium condition) and the energetic and the dissipative stresses are constitutively
related by σ = σ
+ σ
, thus (since here σ
≡ 0)
σ ≡ σ
,
(6.9)
moreover the notions of viscous overstress and plastic stress defined as the values
σ := σ v + σ p with σ v := σ
v and σ p := σ
p
(6.10)
will exclusively be used in the sequel for convenience of exposition.
Separate Viscous Overstress and Plastic Stress
The Bingham 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 union
of the rigid domain and the yield surface, compare the representation in Fig. 5.2.
Thereby, the admissible domain may either be determined directly from the expression of the sub-differential d ˙
π p in Eq. 6.8, or, alternatively, from evaluating the
formal definition of the sub-differential
d ˙
π p (˙ ) = {σ p | σ p [˙
− ˙
] ≤ σ y
|˙
| − |˙ |
∀˙
},
(6.11)
whereby ˙
denotes any admissible strain rate. Then at ˙
= 0 it holds for any admissible ˙
that σ p ˙
≤ σ y |˙
| and, with max ˙
{σ p ˙
/|˙
|} = |σ p |, the admissible domain
follows as |σ p | ≤ σ y .
The rigid domain is defined as the interior of the admissible domain, i.e.
int A := {σ p | |σ p | − σ y < 0},
(6.12)
whereas the yield surface, which in the present one-dimensional case collapses to
the two end points σ p = ±σ y , is defined as the boundary of the admissible domain,
i.e.
∂ A := {σ p | |σ p | − σ y = 0}.
(6.13)
Collectively, the admissible domain in the σ p -space is characterized by the yield
condition
|σ p | − σ y ≤ 0.
(6.14)