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,

$$\eta\Delta{}z=h\dfrac{\Delta{}u}{u_y}\left(A-\nu\left(\gamma+\text{sign}\left(z\cdot\Delta{}u\right)\beta\right) \Big|z\Big|^n\right).$$

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 $u$. 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

vanilla model

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

strength degradation

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

strength degradation

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

stiffness degradation

Pinching Effect

The pinching effect is governed by $\phi_0$, $\delta_\phi$, $\zeta$, $p$, $q$ 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

pinching effect