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

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,

$$\eta \mathrm{\Delta }z=h{\displaystyle \frac{\mathrm{\Delta }u}{u_y}}(A-\nu (\gamma +\text{sign}(z\cdot \mathrm{\Delta }u)\beta )|z{|}^n).$$

Then,

$$F=a{F}_y{\displaystyle \frac{u}{u_y}}+(1-a)F_{y}z.$$

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.

$$e=(1-a)\int z\mathrm{d}u.$$

The trapezoidal rule is used so that

$$\mathrm{\Delta }e=(1-a){\displaystyle \frac{2z+\mathrm{\Delta }z}{2}}\mathrm{\Delta }u.$$

The evolutions are

$$\begin{array}{rl}\nu & ={\nu }_0+{\delta }_{\nu }e,\\ \eta & ={\eta }_0+{\delta }_{\eta }e,\\ A& =1-{\delta }_{A}e,\\ h& =1-{\zeta }_1\exp (-{\left({\displaystyle \frac{z\cdot \text{sign}\left(\mathrm{\Delta }u\right)-q{z}_u}{{\zeta }_2}}\right)}^2),\\ {\zeta }_1& =\zeta (1-\exp (-pe)),\\ {\zeta }_2& =({\varphi }_0+{\delta }_{\varphi }e)(\lambda +{\zeta }_1),\\ z_u& =\sqrt[n]{{\displaystyle \frac{1}{\nu }}}={\nu }^{-1/n}.\end{array}$$

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.

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.

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.

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.

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 ${\varphi }_0$, ${\delta }_{\varphi }$, $\zeta$, $p$, $q$ and $\lambda$.

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