5.3 Prandtl Hardening Model
273
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 5.28b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal with ˙
σ(t) = E ˙
(t) in the elastic phases where |σ(t) −
K κ(t)| = |σ(t) − 0.1 κ(t)| < 1 + 0.1 κ = σ y + H κ(t) (E = 1, thus the slopes in
the elastic phases in Fig. 5.28a, b coincide), and ˙
σ(t) = [E − E
2
/[E + H + K ]] ˙
(t)
= ˙
(t)/6 in the plastic phases where |σ(t) − K κ(t)| = |σ(t) − 0.1 κ(t)| = 1 +
0.1 κ(t) = σ y + H κ(t).
The resulting σ = σ() diagram is highlighted in Fig. 5.28c. Due to the finite
sized time step t and corresponding finite sized strain increment the expected
parallelogram-type format (contracting in the direction and isotropically expanding,
kinematically shifting in the σ direction) of the σ = σ() diagram is only approximately captured in the elastic-plastic transition, however the slopes at = 0 and
= ±5 obviously tend to the elastic modulus E = 1 for t → 0. It is tedious
but easy to verify that the stress varies between [0, 1], [5/3, −1], [ − 20/9, 15/9],
[145/54, −120/54], [ − 995/324, 870/324] and [6595/1944, −3125/1944] in the
elastic phases.
Figure 5.28d demonstrates the plastic strain history p (t): during the plastic
phases p (t) evolves in parallel to the total strain with |˙ p (t)| = |˙ (t)| E/[E +
H + K ] = 5 × 10/12, whereas p (t) stays constant with | p (t)| = {0, 10/3, 25/9,
125/54, 625/324, 3125/1944} during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.28e follows linear in time from integrating ˙
κ(t) = |˙ p (t)| = 5 × 10/12 during the plastic phases and constant in time
during the elastic phases, thus κ max = 2 × [10/3 + 25/9 + 125/54 + 625/324] +
3125/1944 = 43385/1944 ≈ 22.3.
Prescribed Strain History: Sine
The response of the specific Prandtl mixed (isotropic and kinematic) hardening model
to a prescribed Sine strain history is documented in Fig. 5.29a, b, c, d, e.
Figure 5.29a depicts the prescribed Sine strain history (t) = a sin(ω t) with
amplitude a = 5, period T = 4 and corresponding angular frequency ω = 2π/T
in the time interval t ∈ [0, t max = 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 5.29b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal with ˙
σ(t) = ˙
(t) in the elastic phases where |σ(t) −
0.1 κ(t)| < 1 + 0.1 κ(t) (thus the corresponding curve segments representing elastic
loading/unloading in Fig. 5.29a, b are affine), and ˙
σ(t) = ˙
(t)/6 in the plastic phases
where |σ(t) − 0.1 κ(t)| = 1 + 0.1 κ(t).
The resulting σ = σ() diagram is highlighted in Fig. 5.29c. Due to the finite
sized time step t and corresponding finite sized strain increment the expected
parallelogram-type format (contracting in the direction and isotropically expanding,
kinematically shifting in the σ direction) of the σ = σ() diagram is only approx-
273
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 5.28b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal with ˙
σ(t) = E ˙
(t) in the elastic phases where |σ(t) −
K κ(t)| = |σ(t) − 0.1 κ(t)| < 1 + 0.1 κ = σ y + H κ(t) (E = 1, thus the slopes in
the elastic phases in Fig. 5.28a, b coincide), and ˙
σ(t) = [E − E
2
/[E + H + K ]] ˙
(t)
= ˙
(t)/6 in the plastic phases where |σ(t) − K κ(t)| = |σ(t) − 0.1 κ(t)| = 1 +
0.1 κ(t) = σ y + H κ(t).
The resulting σ = σ() diagram is highlighted in Fig. 5.28c. Due to the finite
sized time step t and corresponding finite sized strain increment the expected
parallelogram-type format (contracting in the direction and isotropically expanding,
kinematically shifting in the σ direction) of the σ = σ() diagram is only approximately captured in the elastic-plastic transition, however the slopes at = 0 and
= ±5 obviously tend to the elastic modulus E = 1 for t → 0. It is tedious
but easy to verify that the stress varies between [0, 1], [5/3, −1], [ − 20/9, 15/9],
[145/54, −120/54], [ − 995/324, 870/324] and [6595/1944, −3125/1944] in the
elastic phases.
Figure 5.28d demonstrates the plastic strain history p (t): during the plastic
phases p (t) evolves in parallel to the total strain with |˙ p (t)| = |˙ (t)| E/[E +
H + K ] = 5 × 10/12, whereas p (t) stays constant with | p (t)| = {0, 10/3, 25/9,
125/54, 625/324, 3125/1944} during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.28e follows linear in time from integrating ˙
κ(t) = |˙ p (t)| = 5 × 10/12 during the plastic phases and constant in time
during the elastic phases, thus κ max = 2 × [10/3 + 25/9 + 125/54 + 625/324] +
3125/1944 = 43385/1944 ≈ 22.3.
Prescribed Strain History: Sine
The response of the specific Prandtl mixed (isotropic and kinematic) hardening model
to a prescribed Sine strain history is documented in Fig. 5.29a, b, c, d, e.
Figure 5.29a depicts the prescribed Sine strain history (t) = a sin(ω t) with
amplitude a = 5, period T = 4 and corresponding angular frequency ω = 2π/T
in the time interval t ∈ [0, t max = 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 5.29b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal with ˙
σ(t) = ˙
(t) in the elastic phases where |σ(t) −
0.1 κ(t)| < 1 + 0.1 κ(t) (thus the corresponding curve segments representing elastic
loading/unloading in Fig. 5.29a, b are affine), and ˙
σ(t) = ˙
(t)/6 in the plastic phases
where |σ(t) − 0.1 κ(t)| = 1 + 0.1 κ(t).
The resulting σ = σ() diagram is highlighted in Fig. 5.29c. Due to the finite
sized time step t and corresponding finite sized strain increment the expected
parallelogram-type format (contracting in the direction and isotropically expanding,
kinematically shifting in the σ direction) of the σ = σ() diagram is only approx-