252
5 Plasticity
Table 5.10 Algorithmic update for the specific Prandtl kinematic hardening model
Input
n n−1
p
n−1
hk
Trial Strain
p = n−1
p
hk =
n−1
hk
Trial Stress
σ
p = −E [
p − n ]
σ
hk = −K
hk
Trial Yield
φ = |σ
p + σ
hk | − σ y
Loading Check IF φ < 0 THEN
λ = 0
ELSE
λ =
φ
E + K
ENDIF
Update Strain n
p =
p + λ
σ
p + σ
hk
|σ
p + σ
hk |
n
hk =
hk + λ
σ
p + σ
hk
|σ
p + σ
hk |
Update Stress σ n = E [ n − n
p ]
Tangent
E n
a = E − H 0 ((λ)
E 2
E + K
Output
σ n n
p n
hk E n
a
5.3.6 Specific Prandtl Kinematic Hardening Model:
Response Analysis
Prescribed Strain History: Zig-Zag
The response of the specific Prandtl kinematic hardening model to a prescribed ZigZag strain history is documented in Fig. 5.21a, b, c, d, e.
Figure 5.21a depicts the prescribed Zig-Zag strain history (t) with amplitude
a = 5 and period T = 4 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.