5.3 Prandtl Hardening Model
275
imately 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 5.29d 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 ] = |˙ (t)| × 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. 5.29e follows as a sequence of sine waves
segments from integrating ˙
κ(t) = |˙ p (t)| = |˙ (t)| × 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: Ramp
The response of the specific Prandtl mixed (isotropic and kinematic) hardening model
to a prescribed Ramp strain history is documented in Fig. 5.30a, b, c, d, e.
Figure 5.30a depicts the prescribed Ramp strain history (t) with maximum 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 5.30b showcases the resulting stress history σ(t) with ˙
σ(t) = ˙
(t) in the
two elastic phases where |σ(t) − 0.1 κ(t)| < 1 + 0.1 κ(t) (thus the slopes in the
two elastic phases in Fig. 5.30a, b coincide), and ˙
σ(t) = ˙
(t)/6 in the two plastic
phases where |σ(t) − 0.1 κ(t)| = 1 + 0.1 κ(t). In particular during the holding phase
σ(t) = 5/3 results as the response to the elastic strain e (t) = (t) − p (t) = 5 −
0.8 × 50/12 = 5/3.
The resulting σ = σ() diagram is highlighted in Fig. 5.30c. Due to the finite
sized time step t and corresponding finite sized strain increment the expected
parallelogram-type format (isotropically expanding, kinematically shifting in the σ
direction) of the σ = σ() diagram is only approximately captured in the elasticplastic transition, however the slopes at = 0 and = 5 obviously tend to the elastic
modulus E = 1 for t → 0.
Figure 5.30d 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 ] =
50/12 (or ˙
p (t) = 0 in the holding phase), whereas p (t) stays constant with p (t) =
0.8 × 50/12 = 10/3 (or as initial value p (t) = 0) during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.30e follows constant-linear-constantlinear in time from integrating ˙
κ(t) = |˙ p (t)| = {0, 50/12, 0, 50/12} over the time
interval t ∈ [0, t max = 10], thus κ max = 10/3 + 35/18 = 95/18 ≈ 5.3.
275
imately 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 5.29d 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 ] = |˙ (t)| × 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. 5.29e follows as a sequence of sine waves
segments from integrating ˙
κ(t) = |˙ p (t)| = |˙ (t)| × 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: Ramp
The response of the specific Prandtl mixed (isotropic and kinematic) hardening model
to a prescribed Ramp strain history is documented in Fig. 5.30a, b, c, d, e.
Figure 5.30a depicts the prescribed Ramp strain history (t) with maximum 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 5.30b showcases the resulting stress history σ(t) with ˙
σ(t) = ˙
(t) in the
two elastic phases where |σ(t) − 0.1 κ(t)| < 1 + 0.1 κ(t) (thus the slopes in the
two elastic phases in Fig. 5.30a, b coincide), and ˙
σ(t) = ˙
(t)/6 in the two plastic
phases where |σ(t) − 0.1 κ(t)| = 1 + 0.1 κ(t). In particular during the holding phase
σ(t) = 5/3 results as the response to the elastic strain e (t) = (t) − p (t) = 5 −
0.8 × 50/12 = 5/3.
The resulting σ = σ() diagram is highlighted in Fig. 5.30c. Due to the finite
sized time step t and corresponding finite sized strain increment the expected
parallelogram-type format (isotropically expanding, kinematically shifting in the σ
direction) of the σ = σ() diagram is only approximately captured in the elasticplastic transition, however the slopes at = 0 and = 5 obviously tend to the elastic
modulus E = 1 for t → 0.
Figure 5.30d 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 ] =
50/12 (or ˙
p (t) = 0 in the holding phase), whereas p (t) stays constant with p (t) =
0.8 × 50/12 = 10/3 (or as initial value p (t) = 0) during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.30e follows constant-linear-constantlinear in time from integrating ˙
κ(t) = |˙ p (t)| = {0, 50/12, 0, 50/12} over the time
interval t ∈ [0, t max = 10], thus κ max = 10/3 + 35/18 = 95/18 ≈ 5.3.