298
6 Visco-Plasticity
and the Usawa iteration is continued
2 upon incrementing k and re-computing the
trial total stress σ
. Otherwise, if
n
e falls below the given tolerance, the total stress
is updated as
σ
n
= σ
k−1
vp .
(6.54)
The algorithmic step-by-step update for the specific Bingham model is summarized in Table 6.2.
6.1.3 Specific Bingham Model: Response Analysis
Prescribed Strain History: Zig-Zag
The response of the specific Bingham model to a prescribed Zig-Zag strain history is
documented in Fig. 6.4a–e. (These shall be compared to the corresponding response
of the underlying, viscous and rigid-plastic, specific Newton and St. Venant models
in Figs. 4.6a–f and 5.4a–e, respectively.)
Figure 6.4a 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.
Figure 6.4b showcases the resulting stress history σ(t) that displays a block signal
with σ(t) = ±[σ y + η |˙ (t)|] = ±1.375 whenever ˙
(t) = ±5.
The resulting σ = σ() diagram is highlighted in Fig. 6.4c. Due to the finite sized
time step t and corresponding finite sized strain increment the expected rectangular format of the σ = σ() diagram is only approximately captured, however the
slopes at = 0 and = ±5 obviously tend to ∞ with t → 0.
Figure 6.4d clearly demonstrates that the augmented Lagrange multiplier format
effectively enforces the constraint vp (t) ≡ (t).
Finally, the strain arc-length κ(t) in Fig. 6.4e follows linear in time from integrating ˙
κ(t) = |˙ (t)| = 5 over two and a half periods, thus κ max = 50.
Prescribed Strain History: Sine
The response of the specific Bingham model to a prescribed Sine strain history is
documented in Fig. 6.5a–e. (These shall be compared to the corresponding response
of the underlying, viscous and rigid-plastic, specific Newton and St. Venant models
in Figs. 4.7a–f and 5.5a–e, respectively.)
2 The case of stress control with prescribed σ n .
= ¯
σ n necessitates a special treatment: After resetting
the visco-plastic stress the total strain needs first to be updated according to
n ←
n +
¯
σ n − σ n
E
=
n +
¯
σ n −
σ k
vp − E [ n
vp − n ]
E
=
n
vp +
¯
σ n − σ k
vp
E
before re-computing the trial total stress σ . Thereby it is clearly understood that the penalty parameter E serves as the algorithmic tangent stiffness within the combined augmented Lagrange/Newton
iterations for stress control.
6 Visco-Plasticity
and the Usawa iteration is continued
2 upon incrementing k and re-computing the
trial total stress σ
. Otherwise, if
n
e falls below the given tolerance, the total stress
is updated as
σ
n
= σ
k−1
vp .
(6.54)
The algorithmic step-by-step update for the specific Bingham model is summarized in Table 6.2.
6.1.3 Specific Bingham Model: Response Analysis
Prescribed Strain History: Zig-Zag
The response of the specific Bingham model to a prescribed Zig-Zag strain history is
documented in Fig. 6.4a–e. (These shall be compared to the corresponding response
of the underlying, viscous and rigid-plastic, specific Newton and St. Venant models
in Figs. 4.6a–f and 5.4a–e, respectively.)
Figure 6.4a 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.
Figure 6.4b showcases the resulting stress history σ(t) that displays a block signal
with σ(t) = ±[σ y + η |˙ (t)|] = ±1.375 whenever ˙
(t) = ±5.
The resulting σ = σ() diagram is highlighted in Fig. 6.4c. Due to the finite sized
time step t and corresponding finite sized strain increment the expected rectangular format of the σ = σ() diagram is only approximately captured, however the
slopes at = 0 and = ±5 obviously tend to ∞ with t → 0.
Figure 6.4d clearly demonstrates that the augmented Lagrange multiplier format
effectively enforces the constraint vp (t) ≡ (t).
Finally, the strain arc-length κ(t) in Fig. 6.4e follows linear in time from integrating ˙
κ(t) = |˙ (t)| = 5 over two and a half periods, thus κ max = 50.
Prescribed Strain History: Sine
The response of the specific Bingham model to a prescribed Sine strain history is
documented in Fig. 6.5a–e. (These shall be compared to the corresponding response
of the underlying, viscous and rigid-plastic, specific Newton and St. Venant models
in Figs. 4.7a–f and 5.5a–e, respectively.)
2 The case of stress control with prescribed σ n .
= ¯
σ n necessitates a special treatment: After resetting
the visco-plastic stress the total strain needs first to be updated according to
n ←
n +
¯
σ n − σ n
E
=
n +
¯
σ n −
σ k
vp − E [ n
vp − n ]
E
=
n
vp +
¯
σ n − σ k
vp
E
before re-computing the trial total stress σ . Thereby it is clearly understood that the penalty parameter E serves as the algorithmic tangent stiffness within the combined augmented Lagrange/Newton
iterations for stress control.