5.1 St. Venant Model
195
Thus the energetic stress σ
conjugated to the total strain vanishes identically as
well
σ
() ≡ 0.
(5.3)
Furthermore, for the specific St. Venant model the convex but non-smooth dissipation
potential π is chosen as
π(˙ ) = σ y |˙ |.
(5.4)
Observe that (i) π does depend on ˙
, thus the dissipative stress σ
= 0 for ˙
= 0, and
that (ii) π is positively homogenous of degree one in ˙
and obviously non-smooth
at the origin ˙
= 0. Consequently, the dissipative stress σ
computes as some subderivative of the dissipation potential with respect to its conjugated variable
σ
(˙ ) ∈ d ˙
π(˙ ) =
⎧
⎨
⎩
+σ y
˙
> 0
[ − σ y , +σ y ] for ˙
= 0
−σ y
˙
< 0
⎫
⎬
⎭
,
(5.5)
whereby d ˙
π denotes the set of sub-derivatives, i.e. the sub-differential of π with
respect to ˙
. Observe that σ
is constitutively not determined for ˙
= 0 (thus it can
at best be computed from equilibrium considerations).
Recall that the total stress σ 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 the value (since here σ
≡ 0)
σ ≡ σ
(5.6)
will exclusively be used in the sequel for convenience of exposition.
The closed and convex admissible domain A = int A ∪ ∂ A in the space of the
dissipative driving force, i.e. in the σ-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 in Eq. 5.5, or, alternatively, from evaluating the formal definition
of the sub-differential
d ˙
π(˙ ) = {σ | σ [˙
− ˙
] ≤ σ y
|˙
| − |˙ |
∀˙
},
(5.7)
195
Thus the energetic stress σ
conjugated to the total strain vanishes identically as
well
σ
() ≡ 0.
(5.3)
Furthermore, for the specific St. Venant model the convex but non-smooth dissipation
potential π is chosen as
π(˙ ) = σ y |˙ |.
(5.4)
Observe that (i) π does depend on ˙
, thus the dissipative stress σ
= 0 for ˙
= 0, and
that (ii) π is positively homogenous of degree one in ˙
and obviously non-smooth
at the origin ˙
= 0. Consequently, the dissipative stress σ
computes as some subderivative of the dissipation potential with respect to its conjugated variable
σ
(˙ ) ∈ d ˙
π(˙ ) =
⎧
⎨
⎩
+σ y
˙
> 0
[ − σ y , +σ y ] for ˙
= 0
−σ y
˙
< 0
⎫
⎬
⎭
,
(5.5)
whereby d ˙
π denotes the set of sub-derivatives, i.e. the sub-differential of π with
respect to ˙
. Observe that σ
is constitutively not determined for ˙
= 0 (thus it can
at best be computed from equilibrium considerations).
Recall that the total stress σ 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 the value (since here σ
≡ 0)
σ ≡ σ
(5.6)
will exclusively be used in the sequel for convenience of exposition.
The closed and convex admissible domain A = int A ∪ ∂ A in the space of the
dissipative driving force, i.e. in the σ-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 in Eq. 5.5, or, alternatively, from evaluating the formal definition
of the sub-differential
d ˙
π(˙ ) = {σ | σ [˙
− ˙
] ≤ σ y
|˙
| − |˙ |
∀˙
},
(5.7)