236
5 Plasticity
coincide), and ˙
σ(t) = [E − E
2
/[E + H ]] ˙
(t) = ˙
(t)/11 in the plastic phases where
|σ(t)| = σ y + H κ(t) = 1 + 0.1 κ(t).
The resulting σ = σ() diagram is highlighted in Fig. 5.14c. 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
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], ±15/11, ∓245/11
2
, ±3415/11
3
, ∓44045/11
4 and
[542815/11
5
, −262440/11
5
] in the elastic phases.
Figure 5.14d 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 ] = 5 ×
10/11, whereas p (t) stays constant with | p (t)| = {0, 40/11, 360/11
2
, 3240/11
3
,
29160/11
4
, 262440/11
5
} during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.14e follows linear in time from integrating ˙
κ(t) = |˙ p (t)| = 5 × 10/11 during the plastic phases and constant in time during
the elastic phases, thus κ max = 2 × [40/11 + 360/11
2
+ 3240/11
3
+ 29160/11
4
] +
262440/11
5
= 3817640/11
5
≈ 24.
Prescribed Strain History: Sine
The response of the specific Prandtl isotropic hardening model to a prescribed Sine
strain history is documented in Fig. 5.15a, b, c, d, e.
Figure 5.15a 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 5.15b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal with ˙
σ(t) = ˙
(t) in the elastic phases where |σ(t)| <
1 + 0.1 κ(t) (thus the corresponding curve segments representing elastic loading/unloading in Fig. 5.15a, b are affine), and ˙
σ(t) = ˙
(t)/11 in the plastic phases
where |σ(t)| = 1 + 0.1 κ(t).
The resulting σ = σ() diagram is highlighted in Fig. 5.15c. 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 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], ±15/11, ∓245/11
2
, ±3415/11
3
, ∓44045/11
4 and
[542815/11
5
, −262440/11
5
] in the elastic phases.
Figure 5.15d 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 ] = |˙ (t)| ×
5 Plasticity
coincide), and ˙
σ(t) = [E − E
2
/[E + H ]] ˙
(t) = ˙
(t)/11 in the plastic phases where
|σ(t)| = σ y + H κ(t) = 1 + 0.1 κ(t).
The resulting σ = σ() diagram is highlighted in Fig. 5.14c. 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
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], ±15/11, ∓245/11
2
, ±3415/11
3
, ∓44045/11
4 and
[542815/11
5
, −262440/11
5
] in the elastic phases.
Figure 5.14d 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 ] = 5 ×
10/11, whereas p (t) stays constant with | p (t)| = {0, 40/11, 360/11
2
, 3240/11
3
,
29160/11
4
, 262440/11
5
} during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.14e follows linear in time from integrating ˙
κ(t) = |˙ p (t)| = 5 × 10/11 during the plastic phases and constant in time during
the elastic phases, thus κ max = 2 × [40/11 + 360/11
2
+ 3240/11
3
+ 29160/11
4
] +
262440/11
5
= 3817640/11
5
≈ 24.
Prescribed Strain History: Sine
The response of the specific Prandtl isotropic hardening model to a prescribed Sine
strain history is documented in Fig. 5.15a, b, c, d, e.
Figure 5.15a 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 5.15b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal with ˙
σ(t) = ˙
(t) in the elastic phases where |σ(t)| <
1 + 0.1 κ(t) (thus the corresponding curve segments representing elastic loading/unloading in Fig. 5.15a, b are affine), and ˙
σ(t) = ˙
(t)/11 in the plastic phases
where |σ(t)| = 1 + 0.1 κ(t).
The resulting σ = σ() diagram is highlighted in Fig. 5.15c. 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 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], ±15/11, ∓245/11
2
, ±3415/11
3
, ∓44045/11
4 and
[542815/11
5
, −262440/11
5
] in the elastic phases.
Figure 5.15d 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 ] = |˙ (t)| ×