376
6 Visco-Plasticity
with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger
hollow circles, whereas elastic time steps are indicated by smaller filled circles.
Figure 6.32b showcases the resulting strain history (t) that displays a smoothly
increasing signal in the loading and holding phases with (t) → 45 and a smoothly
decreasing signal in the unloading phase with (t) → 30 (from visual inspection).
The nonlinear creep during the holding phase saturates due to the kinematic hardening.
The resulting σ = σ() diagram is highlighted in Fig. 6.32c. Once the holding
phase is completed the σ = σ() behavior in the unloading phase is initially purely
elastic with σ(t) ∈ [5, 3] and slope E = 1, and subsequently visco-plastic whereby
the strain approaches (t) → 30. Likewise the creep towards = 45 during the holding phase is clearly visible at σ = 5.
Figure 6.32d demonstrates the corresponding smoothly and monotonically in/decreasing visco-plastic strain history vp (t) with vp (t) → 40 during the holding
phase and vp (t) → 30 in the unloading phase.
Finally, the strain arc-length κ(t) in Fig. 6.32e follows from integrating ˙
κ(t) =
|˙ (t)| over the time interval t ∈ [0, t max = 10] and approaches κ max ≈ 50 (from visual
inspection).
6.3.7 Specific Perzyna Mixed Hardening Model: Formulation
The specific Perzyna (isotropic and kinematic) mixed hardening model, similar to
that displayed in Fig. 6.18 (however with the hardening modulus H and the hardening strain ε h coinciding here with the isotropic- and kinematic-hardening moduli
H and K , respectively, and the isotropic- and kinematic-hardening strains hi and
hk , respectively), consists of a serial arrangement of (1) a linear elastic spring with
stiffness E and (2) a linear mixed-hardening viscous frictional slider consisting of
a parallel arrangement of (i) a linear frictional slider with threshold σ y , (ii) a linear viscous dashpot with viscosity η, and (iii) linear mixed-hardening springs with
stiffnesses H and K (the isotropic- and kinematic-hardening moduli).
For the specific Perzyna mixed hardening model the free energy density ψ is
expressed as a quadratic (and thus convex) function of − vp (the elastic strain e ),
hi (the isotropic-hardening strain) and hk (the kinematic-hardening strain)
ψ(, vp , hi , hk ) =
1
2
E [ − vp ]
2
+
1
2
H
2
hi +
1
2
K
2
hk .
(6.210)
Then the energetic stress σ
conjugated to the total strain and the energetic
visco-plastic stress σ
vp conjugated to the visco-plastic strain vp together with the
isotropic-hardening stress σ
hi conjugated to the isotropic-hardening strain
hi and
the kinematic-hardening stress σ
hk conjugated to the kinematic-hardening strain
hk
follow as
6 Visco-Plasticity
with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger
hollow circles, whereas elastic time steps are indicated by smaller filled circles.
Figure 6.32b showcases the resulting strain history (t) that displays a smoothly
increasing signal in the loading and holding phases with (t) → 45 and a smoothly
decreasing signal in the unloading phase with (t) → 30 (from visual inspection).
The nonlinear creep during the holding phase saturates due to the kinematic hardening.
The resulting σ = σ() diagram is highlighted in Fig. 6.32c. Once the holding
phase is completed the σ = σ() behavior in the unloading phase is initially purely
elastic with σ(t) ∈ [5, 3] and slope E = 1, and subsequently visco-plastic whereby
the strain approaches (t) → 30. Likewise the creep towards = 45 during the holding phase is clearly visible at σ = 5.
Figure 6.32d demonstrates the corresponding smoothly and monotonically in/decreasing visco-plastic strain history vp (t) with vp (t) → 40 during the holding
phase and vp (t) → 30 in the unloading phase.
Finally, the strain arc-length κ(t) in Fig. 6.32e follows from integrating ˙
κ(t) =
|˙ (t)| over the time interval t ∈ [0, t max = 10] and approaches κ max ≈ 50 (from visual
inspection).
6.3.7 Specific Perzyna Mixed Hardening Model: Formulation
The specific Perzyna (isotropic and kinematic) mixed hardening model, similar to
that displayed in Fig. 6.18 (however with the hardening modulus H and the hardening strain ε h coinciding here with the isotropic- and kinematic-hardening moduli
H and K , respectively, and the isotropic- and kinematic-hardening strains hi and
hk , respectively), consists of a serial arrangement of (1) a linear elastic spring with
stiffness E and (2) a linear mixed-hardening viscous frictional slider consisting of
a parallel arrangement of (i) a linear frictional slider with threshold σ y , (ii) a linear viscous dashpot with viscosity η, and (iii) linear mixed-hardening springs with
stiffnesses H and K (the isotropic- and kinematic-hardening moduli).
For the specific Perzyna mixed hardening model the free energy density ψ is
expressed as a quadratic (and thus convex) function of − vp (the elastic strain e ),
hi (the isotropic-hardening strain) and hk (the kinematic-hardening strain)
ψ(, vp , hi , hk ) =
1
2
E [ − vp ]
2
+
1
2
H
2
hi +
1
2
K
2
hk .
(6.210)
Then the energetic stress σ
conjugated to the total strain and the energetic
visco-plastic stress σ
vp conjugated to the visco-plastic strain vp together with the
isotropic-hardening stress σ
hi conjugated to the isotropic-hardening strain
hi and
the kinematic-hardening stress σ
hk conjugated to the kinematic-hardening strain
hk
follow as