5.3 Prandtl Hardening Model
243
Prescribed Stress History: Ramp
The response of the specific Prandtl isotropic hardening model to a prescribed Ramp
stress history is documented in Fig. 5.19a, b, c, d, e.
Figure 5.19a depicts the prescribed Ramp stress history σ(t) with maximum σ a =
5, loading phase during t ∈ [t 0 = 0, t 1 = 1), holding phase during t ∈ [t 1 = 1, t 2 =
9], and unloading phase during t ∈ (t 2 = 9, t 3 = 10], whereby N = 100 time steps
with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow
circles, whereas elastic time steps are indicated by smaller filled circles.
Figure 5.19b showcases the resulting strain history (t) that displays an initial
elastic phase with ˙
(t) = ˙
σ(t), a subsequent plastic phase with ˙
(t) = ˙
σ(t) × 11
(thus in particular during the holding phase (t) = 1/1 + 4 × 11 = 45), and a final
elastic phase with ˙
(t) = ˙
σ(t), respectively.
The resulting σ = σ() diagram is highlighted in Fig. 5.19c. Once the holding
phase is completed the σ = σ() behavior in the unloading phase is linear elastic
with slope E = 1. It is easy to verify that the strain varies between [45, 40] in the
final elastic phase.
Figure 5.19d demonstrates the plastic strain history p (t): during the plastic phase
p (t) evolves with |˙ p (t)| = |˙ (t) − ˙
σ(t)/E| = 55 − 5/1 = 50, whereas p (t) stays
constant with p (t) = 40 (or as initial value p (t) = 0) during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.19e follows constant-linear-constant in
time from integrating ˙
κ(t) = |˙ p (t)| = {0, 50, 0} over the time interval t ∈ [0, t max =
10], thus κ max = 40.
5.3.4 Specific Prandtl Kinematic Hardening Model:
Formulation
The specific Prandtl kinematic hardening model, similar to that displayed in Fig. 5.12
(however with the hardening modulus H and the hardening strain ε h coinciding here
with the kinematic-hardening modulus K and the kinematic-hardening strain hk ,
respectively), consists of a serial arrangement of (1) a linear elastic spring with
stiffness E and (2) a linear kinematic-hardening frictional slider consisting of a
parallel arrangement of (i) a linear frictional slider with threshold σ y and (ii) a linear
kinematic-hardening spring with stiffness K (the kinematic-hardening modulus).
For the specific Prandtl kinematic hardening model the free energy density ψ is
expressed as a quadratic (and thus convex) function of − p (i.e. the elastic strain
e ) and hk (the kinematic-hardening strain)
ψ(, p , hk ) =
1
2
E [ − p ]
2
+
1
2
K
2
hk .
(5.140)
Then the energetic stress σ
conjugated to the total strain and the energetic plastic
stress σ
p conjugated to the plastic strain p together with the kinematic-hardening
243
Prescribed Stress History: Ramp
The response of the specific Prandtl isotropic hardening model to a prescribed Ramp
stress history is documented in Fig. 5.19a, b, c, d, e.
Figure 5.19a depicts the prescribed Ramp stress history σ(t) with maximum σ a =
5, loading phase during t ∈ [t 0 = 0, t 1 = 1), holding phase during t ∈ [t 1 = 1, t 2 =
9], and unloading phase during t ∈ (t 2 = 9, t 3 = 10], whereby N = 100 time steps
with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow
circles, whereas elastic time steps are indicated by smaller filled circles.
Figure 5.19b showcases the resulting strain history (t) that displays an initial
elastic phase with ˙
(t) = ˙
σ(t), a subsequent plastic phase with ˙
(t) = ˙
σ(t) × 11
(thus in particular during the holding phase (t) = 1/1 + 4 × 11 = 45), and a final
elastic phase with ˙
(t) = ˙
σ(t), respectively.
The resulting σ = σ() diagram is highlighted in Fig. 5.19c. Once the holding
phase is completed the σ = σ() behavior in the unloading phase is linear elastic
with slope E = 1. It is easy to verify that the strain varies between [45, 40] in the
final elastic phase.
Figure 5.19d demonstrates the plastic strain history p (t): during the plastic phase
p (t) evolves with |˙ p (t)| = |˙ (t) − ˙
σ(t)/E| = 55 − 5/1 = 50, whereas p (t) stays
constant with p (t) = 40 (or as initial value p (t) = 0) during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.19e follows constant-linear-constant in
time from integrating ˙
κ(t) = |˙ p (t)| = {0, 50, 0} over the time interval t ∈ [0, t max =
10], thus κ max = 40.
5.3.4 Specific Prandtl Kinematic Hardening Model:
Formulation
The specific Prandtl kinematic hardening model, similar to that displayed in Fig. 5.12
(however with the hardening modulus H and the hardening strain ε h coinciding here
with the kinematic-hardening modulus K and the kinematic-hardening strain hk ,
respectively), consists of a serial arrangement of (1) a linear elastic spring with
stiffness E and (2) a linear kinematic-hardening frictional slider consisting of a
parallel arrangement of (i) a linear frictional slider with threshold σ y and (ii) a linear
kinematic-hardening spring with stiffness K (the kinematic-hardening modulus).
For the specific Prandtl kinematic hardening model the free energy density ψ is
expressed as a quadratic (and thus convex) function of − p (i.e. the elastic strain
e ) and hk (the kinematic-hardening strain)
ψ(, p , hk ) =
1
2
E [ − p ]
2
+
1
2
K
2
hk .
(5.140)
Then the energetic stress σ
conjugated to the total strain and the energetic plastic
stress σ
p conjugated to the plastic strain p together with the kinematic-hardening