264
5 Plasticity
H
H
hi
σ p
0
+σ y
σ p = +σ y +
hk +
hi
−σ y
σ p = −σ y +
hk −
hi
K
K
hk
σ p
0
+σ y
−σ y
Fig. 5.27 Specific Prandtl (isotropic and kinematic) mixed hardening model: The elastic domain
for σ p defined by |σ p − K hk | − [σ y + H hi ] < 0 in the {σ p , hi , hk }-space expands uniformly
with the isotropic hardening strain hi ∈ [0, ∞) and shifts with the kinematic hardening strain
hk ∈ (−∞, +∞). The slopes of the two lines |σ p | − [σ y + H hi ] = 0 defining the yield surface for
hk = 0 denote the isotropic hardening modulus H . The slope of the two lines |σ p − K hk | − σ y =
0 defining the yield surface for hi = 0 denotes the kinematic hardening modulus K . The union of the
elastic domain and the yield surface renders the admissible domain |σ p − K hk | − [σ y + H hi ] ≤ 0
σ p := σ
p = −σ
p ,
(5.180a)
σ hi := σ
hi = −σ
hi ,
(5.180b)
σ hk := σ
hk = −σ
hk ,
(5.180c)
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 forces, i.e. in the {σ p , σ hi , σ hk }-space, is introduced as the union
of the elastic domain and the yield surface, compare the representation in Fig. 5.27.
Thereby, the admissible domain may either be determined directly from the expression of the sub-differential d ˙
p π in Eq. 5.179, or, alternatively, from evaluating the
formal definition of the sub-differential
d ˙
p π(˙ p , ˙
hi , ˙
hk ) =
(5.181)
{σ p | σ p [˙
p − ˙
p ] ≤ [σ y + H hi ]
|˙
p | − |˙ p |
+ K hk [˙
p − ˙
p ] ∀˙
p },
whereby ˙
p denotes any admissible plastic strain rate. Then at ˙
p = 0 it holds for
any admissible ˙
p that σ p ˙
p ≤ [σ y + H hi ] |˙
p | + K hk ˙
p and, with max ˙
p
{[σ p −
K hk ] ˙
p /|˙
p |} = |σ p − K hk |, the admissible domain follows as |σ p − K hk | ≤
σ y + H hi . Moreover, the sub-differentials d ˙
hi π and d ˙
hk π reduce to the partial derivatives ∂ ˙
hi π and ∂ ˙
hk π, respectively, and render σ hi = −H hi and σ hk = −K hk . Thus
the admissible domain is eventually expressed as |σ p + σ hk | ≤ σ y − σ hi .
The elastic domain is defined as the interior of the admissible domain, i.e.
5 Plasticity
H
H
hi
σ p
0
+σ y
σ p = +σ y +
hk +
hi
−σ y
σ p = −σ y +
hk −
hi
K
K
hk
σ p
0
+σ y
−σ y
Fig. 5.27 Specific Prandtl (isotropic and kinematic) mixed hardening model: The elastic domain
for σ p defined by |σ p − K hk | − [σ y + H hi ] < 0 in the {σ p , hi , hk }-space expands uniformly
with the isotropic hardening strain hi ∈ [0, ∞) and shifts with the kinematic hardening strain
hk ∈ (−∞, +∞). The slopes of the two lines |σ p | − [σ y + H hi ] = 0 defining the yield surface for
hk = 0 denote the isotropic hardening modulus H . The slope of the two lines |σ p − K hk | − σ y =
0 defining the yield surface for hi = 0 denotes the kinematic hardening modulus K . The union of the
elastic domain and the yield surface renders the admissible domain |σ p − K hk | − [σ y + H hi ] ≤ 0
σ p := σ
p = −σ
p ,
(5.180a)
σ hi := σ
hi = −σ
hi ,
(5.180b)
σ hk := σ
hk = −σ
hk ,
(5.180c)
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 forces, i.e. in the {σ p , σ hi , σ hk }-space, is introduced as the union
of the elastic domain and the yield surface, compare the representation in Fig. 5.27.
Thereby, the admissible domain may either be determined directly from the expression of the sub-differential d ˙
p π in Eq. 5.179, or, alternatively, from evaluating the
formal definition of the sub-differential
d ˙
p π(˙ p , ˙
hi , ˙
hk ) =
(5.181)
{σ p | σ p [˙
p − ˙
p ] ≤ [σ y + H hi ]
|˙
p | − |˙ p |
+ K hk [˙
p − ˙
p ] ∀˙
p },
whereby ˙
p denotes any admissible plastic strain rate. Then at ˙
p = 0 it holds for
any admissible ˙
p that σ p ˙
p ≤ [σ y + H hi ] |˙
p | + K hk ˙
p and, with max ˙
p
{[σ p −
K hk ] ˙
p /|˙
p |} = |σ p − K hk |, the admissible domain follows as |σ p − K hk | ≤
σ y + H hi . Moreover, the sub-differentials d ˙
hi π and d ˙
hk π reduce to the partial derivatives ∂ ˙
hi π and ∂ ˙
hk π, respectively, and render σ hi = −H hi and σ hk = −K hk . Thus
the admissible domain is eventually expressed as |σ p + σ hk | ≤ σ y − σ hi .
The elastic domain is defined as the interior of the admissible domain, i.e.