BWBN
Bouc-Wen-Baber-Noori Model
The BWBN model is an extension of the BoucWen model with stiffness degradation, strength degradation and pinching
effect.
Syntax
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|---|
| material BWBN (1) [2...18]
# (1) int, unique material tag
# [2] double, elastic modulus, default: 2E5
# [3] double, yield stress, default: 4E2
# [4] double, hardening ratio, default: 1E-2
# [5] double, \beta (>0), default: 0.5
# [6] double, exponent n (>0, normally >=1), default: 1.0
# [7] double, initial \nu (>0), default: 1.0
# [8] double, slope of \nu (>0), default: 0.0
# [9] double, initial \eta (>0), default: 1.0
# [10] double, slope of \eta (>0), default: 0.0
# [11] double, initial \phi (>0), default: 1.0
# [12] double, slope of \phi (>0), default: 0.0
# [13] double, \zeta (1>\zeta>0), default: 0.0
# [14] double, slope of A (>0), default: 0.0
# [15] double, p (>0), default: 0.0
# [16] double, q (>0), default: 0.0
# [17] double, \lambda (>0), default: 1.0
# [18] double, density, default: 0.0
|
History Variable Layout
| location |
value |
initial_history(0) |
z |
Theory
The Wikipedia page contains sufficient information about
the formulation of BWBN model. Some normalizations are carried out compared to the original model.
The evolution of internal displacement \(z(t)\) is governed by the differential equation,
\[
\eta\Delta{}z=h\dfrac{\Delta{}u}{u_y}\left(A-\nu\left(\gamma+\text{sign}\left(z\cdot\Delta{}u\right)\beta\right)
\Big|z\Big|^n\right).
\]
Then,
\[
F=aF_y\dfrac{u}{u_y}+\left(1-a\right)F_yz.
\]
For state determination, \(z\) is solved iteratively by using the Newton method. The evolutions of internal functions
rely on the dissipated energy \(e\), which is defined to be a normalized quantity.
\[
e=\left(1-a\right)\int{}z~\mathrm{d}u.
\]
The trapezoidal rule is used so that
\[
\Delta{}e=\left(1-a\right)\dfrac{2z+\Delta{}z}{2}\Delta{}u.
\]
The evolutions are
\[
\begin{align*} \nu&=\nu_0+\delta_\nu{}e,\\[3mm]
\eta&=\eta_0+\delta_\eta{}e,\\[3mm]
A&=1-\delta_Ae,\\[3mm]
h&=1-\zeta_1\exp\left(-\left(\dfrac{z\cdot\text{sign}\left(\Delta{}u\right)-qz_u}{\zeta_2}\right)^2\right),\\[3mm]
\zeta_1&=\zeta\left(1-\exp\left(-pe\right)\right),\\[3mm]
\zeta_2&=\left(\phi_0+\delta_\phi{}e\right)\left(\lambda+\zeta_1\right),\\[3mm]
z_u&=\sqrt[n]{\dfrac{1}{\nu}}=\nu^{-1/n}. \end{align*}
\]
Parameters
Strength degradation is controlled by \(\nu\). To disable it, set \(\delta_\nu=0\).
Stiffness degradation is controlled by \(\eta\). To disable it, set \(\delta_\eta=0\).
Pinching is controlled by \(h\). To disable it, set \(\zeta=0\) or \(p=0\).
Examples
Vanilla Model
The default behavior is similar to a bilinear hardening material.
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| material BWBN 1
materialtest1d 1 4E-4 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 80
|
Strength degradation
A positive \(\delta_\nu\) enables strength degradation.
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| material BWBN 1 2E5 4E2 0 .5 1. 1. 1E0 1. 0. 1. 0. 0. 0. 0. 0. 1. 0.
materialtest1d 1 4E-4 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 80
|
A positive \(\delta_A\) has the similar effect.
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| material BWBN 1 2E5 4E2 0 .5 1. 1. 0. 1. 0. 1. 0. 0. 1E0 0. 0. 1. 0.
materialtest1d 1 4E-4 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 80
|
Stiffness degradation
A positive \(\delta_\eta\) enables stiffness degradation.
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| material BWBN 1 2E5 4E2 0 .5 1 1 0 1 1E1 1 0 1 0 1 1 0 0
materialtest1d 1 4E-4 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 80
|
Pinching Effect
The pinching effect is governed by \(\phi_0\), \(\delta_\phi\), \(\zeta\), \(p\), \(q\) and \(\lambda\).
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| material BWBN 1 2E5 4E2 0 .5 1. 1. 0. 1. 0. 1. 1E1 1. 0. 1E1 1E0 1. 0.
materialtest1d 1 1E-3 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 80
|
