212
5 Plasticity
Note that the total stress σ applied to the rheological model (that enters the equilibrium
condition) coincides identically with the energetic stress, σ
≡ σ, and, due to the
serial arrangement of the elastic spring and the frictional slider, also with the negative
of the energetic plastic stress, −σ
p ≡ σ.
Furthermore, for the specific Prandtl model the convex but non-smooth dissipation
potential π is chosen as
π(˙ p ) = σ y |˙ p |.
(5.60)
Observe that (i) π does not depend on ˙
, thus the dissipative stress σ
= σ − σ
≡ 0
vanishes identically, and that (ii) π is positively homogenous of degree one in ˙
p
and obviously non-smooth at the origin ˙
p = 0. Consequently, the dissipative plastic
stress σ
p computes as some sub-derivative of the dissipation potential with respect
to its conjugated variable
σ
p (˙ p ) ∈ d ˙
p π(˙ p ) =
⎧
⎨
⎩
+σ y
˙
p > 0
[ − σ y , +σ y ] for ˙
p = 0
−σ y
˙
p < 0
⎫
⎬
⎭
,
(5.61)
whereby d ˙
p π denotes the set of sub-derivatives, i.e. the sub-differential of π with
respect to ˙
p .
Recall that the energetic and the dissipative plastic stresses are constitutively
related by σ
p + σ
p = 0, thus the notion of plastic stress defined as the value
σ p := σ
p = −σ
p
(5.62)
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 p -space, is next introduced as the union of the
elastic domain and the yield surface, compare the representation in Fig. 5.8. Thereby,
the admissible domain may either be determined directly from the expression of
the sub-differential d ˙
p π in Eq. 5.61, or, alternatively, from evaluating the formal
definition of the sub-differential
d ˙
p π(˙ p ) = {σ p | σ p [˙
p − ˙
p ] ≤ σ y
|˙
p | − |˙ p |
∀˙
p },
(5.63)
whereby ˙
p denotes any admissible plastic strain rate. Then at ˙
p = 0 it holds for any
admissible ˙
p that σ p ˙
p ≤ σ y |˙
p | and, with max ˙
p
{σ p ˙
p /|˙
p |} = |σ p |, the admissible
domain follows as |σ p | ≤ σ y .
5 Plasticity
Note that the total stress σ applied to the rheological model (that enters the equilibrium
condition) coincides identically with the energetic stress, σ
≡ σ, and, due to the
serial arrangement of the elastic spring and the frictional slider, also with the negative
of the energetic plastic stress, −σ
p ≡ σ.
Furthermore, for the specific Prandtl model the convex but non-smooth dissipation
potential π is chosen as
π(˙ p ) = σ y |˙ p |.
(5.60)
Observe that (i) π does not depend on ˙
, thus the dissipative stress σ
= σ − σ
≡ 0
vanishes identically, and that (ii) π is positively homogenous of degree one in ˙
p
and obviously non-smooth at the origin ˙
p = 0. Consequently, the dissipative plastic
stress σ
p computes as some sub-derivative of the dissipation potential with respect
to its conjugated variable
σ
p (˙ p ) ∈ d ˙
p π(˙ p ) =
⎧
⎨
⎩
+σ y
˙
p > 0
[ − σ y , +σ y ] for ˙
p = 0
−σ y
˙
p < 0
⎫
⎬
⎭
,
(5.61)
whereby d ˙
p π denotes the set of sub-derivatives, i.e. the sub-differential of π with
respect to ˙
p .
Recall that the energetic and the dissipative plastic stresses are constitutively
related by σ
p + σ
p = 0, thus the notion of plastic stress defined as the value
σ p := σ
p = −σ
p
(5.62)
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 p -space, is next introduced as the union of the
elastic domain and the yield surface, compare the representation in Fig. 5.8. Thereby,
the admissible domain may either be determined directly from the expression of
the sub-differential d ˙
p π in Eq. 5.61, or, alternatively, from evaluating the formal
definition of the sub-differential
d ˙
p π(˙ p ) = {σ p | σ p [˙
p − ˙
p ] ≤ σ y
|˙
p | − |˙ p |
∀˙
p },
(5.63)
whereby ˙
p denotes any admissible plastic strain rate. Then at ˙
p = 0 it holds for any
admissible ˙
p that σ p ˙
p ≤ σ y |˙
p | and, with max ˙
p
{σ p ˙
p /|˙
p |} = |σ p |, the admissible
domain follows as |σ p | ≤ σ y .